old exams and homework

MATHEMATICS ASSIGNMENTS AND

EXAMINATIONS

Elementary Real Analysis

Various years

MATHEMATICS 242 December 7 1987

1. Prove that a sequence of real numbers is convergent if and only if it is aCauchy sequence.

2. Define a sequence of real numbers recursively by writing x0 = a and

xn+1 =

√ab2 + xn

2

a + 1.

Prove that if 0 < a < b then the sequence {xn} is convergent and obtainits limit.

3. Suppose that the series of positive terms∑∞

i=1 ai diverges. What, if any-thing, can be concluded about the following series:

(a)∞∑

i=1

ai/i

(b)∞∑

i=1

ai2

(c)∞∑

i=1

√ai

(d)∞∑

i=1

ai

ai + 1

1

4. Define what is meant by a closed set of real numbers and an open set ofreal numbers. Prove that for a continuous function f on the real line theset of points

{x : 0 < f(x) < 1}is open and that the set of points

{x : 0 ≤ f(x) ≤ 1}

is closed.

5. Define uniform continuity.

(a) Give an example of a function which is continuous but not uniformlycontinuous on the interval (−1, 1). Prove that your example is notuniformly continuous.

(b) Prove directly from your definition of uniform continuity that thefunction f(x) = x3 is uniformly continuous on (−1, 1).

(c) Prove directly from your definition of uniform continuity that if afunction f is uniformly continuous on (−1, 1) then it must be boundedthere.

6. Give an example of a set E (other than ∅ and R) that has the followingproperty or else show that such a set cannot exist:

(a) E has infinitely many points but no points of accumulation.

(b) E has infinitely many points of accumulation but no interior points.

(c) E is open but has no points of accumulation.

(d) E is infinite and bounded but has no points of accumulation.

(e) E is open and unbounded.

2

MATHEMATICS 242

First Assignment

Due date to be determined.

The text for the course is Elementary Mathematical Analysis by C. Clark(Wadsworth, 2nd Ed.). The marking scheme is based on two midterm examina-tions (20% each), weekly assignments (10%) and a three hour final (50%). Theinstructor is B. S. Thomson (TLX 10519, 291-4233).

Construct proofs for the following in the style demanded; in each case it ismore the presentation and style of the proof that we wish to examine. Correctmathematical ideas however are not discouraged.

1. (Direct proof) For all real numbers x and y, if x < y then x3 < y3.

2. (Contraposition) If x is irrational then x + r is irrational for all rationalnumbers r.

3. (Indirect proof) The number 3√

2 is irrational.

4. (Counterexample) For any natural number n the equation 4x2 + x − n = 0has no rational root (?).

5. (Induction) For every n = 1, 2, 3, . . . ,

dn

dxne2x = e2x+n log 2.

3

MATHEMATICS 242

Second Assignment

Due Friday, January 25 1992

1. Prove or disprove each of the following assertions:

• Every monotonic sequence converges.

• Every bounded sequence converges.

• If {sn} is bounded then so too is the sequence {|sn|}.• If {sn} is bounded then so too is the sequence {s2

n}.• If {sn} is monotonic then so too is the sequence {s2

n}.• If {sn} is a monotonic sequence of positive terms then so too is the se-

quence {1/sn}.• If {sn} is bounded but not monotonic then it diverges.

2. Give a complete proof for the statement that

lim1sn

=1

lim sn

for a sequence {sn} under some appropriate hypotheses. (Include an exampleto show that your hypotheses cannot be dropped.)

3. A sequence {xn} is defined by starting at some value x1 = θ and continuinginductively

xn+1 =x2

n − 2xn + 32

.

Discuss the convergence behaviour. (Hint: the cases θ < 1, 1 < θ < 3, etc.should be considered separately.)

4

MATHEMATICS 242

Third Assignment

Due 31 January 1992

1. Give an example of a sequence {sn} having the stated property or else explainbriefly (using appropriate theorems) why no such example can exist.

(i) Each sn < 0 and {sn} has no convergent subsequence.

(ii) Each −2 < sn < 0 and {sn} has no convergent subsequence.

(iii) The sequence {sn} diverges but every subsequence converges.

(iv) The sequence {sn} diverges and every subsequence diverges.

(v) The sequence {sn} converges but every subsequence diverges.

(vi) The sequence {sn} diverges and every subsequence either diverges or elseconverges to 2.

2. Prove (or disprove) directly from the definition of a Cauchy sequence:

(i) Every bounded sequence is Cauchy.

(ii) If {sn} is Cauchy then so too is {|sn|}.

(iii) If {|sn|} is Cauchy then so too is {sn}.

5

3. If x2k → L and x2k+1 → L as k →∞ then xk → L.


xn+1 =

√ab2 + xn

2

a + 1.

Prove that if 0 < a < b then the sequence {xn} is convergent and obtain itslimit.

6

MATHEMATICS 242

Fourth Assignment

Due 7 February 1992

3. A sequence {xn} is said to be contractive if there is some constant α < 1 sothat

|xn+2 − xn+1| ≤ α|xn+1 − xn|for all n. Show that every contractive sequence is convergent. Give an exampleof a contractive sequence. Give an example of a non-contractive sequence. Isevery convergent sequence contractive?

2. Which of these converge

∞∑n=1

n(n + 1)(n + 2)2

,

∞∑n=1

3n(n + 1)(n + 2)n3√

n,

∞∑n=2

1ns log n

∞∑n=1

1.3. . . . (2n− 1)2.4. . . . 2n

√n

∞∑n=1

a1/n − 1,

∞∑n=2

1n(log n)t

∞∑n=2

1ns(log n)t

,

∞∑n=1

(1− 1

n

)n2

3. If∑∞

n=1 an diverges and all an > 0 then what (if anything) can be said about

∞∑n=1

an

1 + an,

∞∑n=1

an

1 + nan,

∞∑n=1

an

1 + n2an?

7

MATHEMATICS 242

Fifth Assignment

Due 14 February 1992

1. Obtain the identity

N∑n=1

1n(n + 1)(n + 2)

=14− 1

2(N + 1)(N + 2).

What isN∑

n=1

1n(n + 1)(n + 2)

?

2. Suppose that∑∞

n=1 an is a convergent series of positive terms. Show that∑∞n=1 a2

n is convergent.

3. Let 0 < a ≤ b and consider the series

a + ab + a2b + a2b2 + a3b2 + a3b3 + . . . .

Show that if a ≥ 1 then the series diverges while if b < 1 the series converges.In general when does the series converge?

4. If the sequence {xn} is monotonically decreasing to zero then the series

x1 − 12(x1 + x2) +

13(x1 + x2 + x3)− 1

4(x1 + x2 + x3 + x4) . . .

converges.

8

MATHEMATICS 242

Sixth Assignment


1. A set E is said to be dense if E = R.

1. Find a set so that E and R \ E are both dense.

2. Find a countable dense set.

3. Find an uncountable dense set.

4. Show that E is dense if and only if E ∩ I 6= ∅ for every open interval I.

5. Show that the intersection of two dense sets need not be dense.

6. Show that the intersection of two dense open sets is dense.

7. Show that the union of two dense sets is dense.


1. E has infinitely many points but no interior points.

2. E has infinitely many points but no points of accumulation.

3. E is open and unbounded.

4. E is closed and unbounded.

5. E has infinitely many points of accumulation but no interior points.

6. E is open but has no points of accumulation.

7. E is closed but has no points of accumulation.

8. E is compact and has no interior points.

9. E, E′ and E′′ are different.

10. E is countable and E′ = {0, 1}.11. E is countable and E′ = [0, 1].

12. E is countable and E′ = (0, 1).

3. Prove that the following definitions for “accumulation point” are equivalent:

9

1. A point x0 is an accumulation point of a set A provided that every deletedneighbourhood of x0 contains some point of A.

2. A point x0 is an accumulation point of a set A provided that every neigh-bourhood of x0 contains two points of A.

3. A point x0 is an accumulation point of a set A provided that every neigh-bourhood of x0 contains infinitely many points of A.

4. A point x0 is an accumulation point of a set A provided that there is asequence {xn} of distinct points of A such that limn→∞ xn = x0.

4. For any set S we let S′ denote the set of its accumulation points. Give anexample that illustrates how each of the following can occur:

1. S′ = Ø.

2. S′ contains just one point.

3. S′ contains exactly two points.

4. S′ is countably infinite.

5. S′ is uncountable.

6. S′ is nonempty but (S′)′= Ø.

10

MATHEMATICS 242

Seventh Assignment

Due 6 March 1992

1. Show that the set of accumulation points of any set is closed.

2. Let {xn} be a convergent sequence and let E be the range of the sequence.What is the closure of E?

3. Let E be a bounded, nonempty open set. Show that supE and inf E arepoints of accumulation of E neither of which belongs to E.

4. The distance between a point x and a nonempty set E is defined to be

d(E, x) = inf{|x− y| : y ∈ E}.

Show that, for a nonempty, closed set E, x ∈ E if and only if d(E, x) = 0. Isthis true if E is not closed?

5. Let E be a closed set. Show that the function f(x) = d(E, x) is a nonnegativefunction which vanishes (i.e. equals zero) only at points of E. For the followingexamples attempt a representation of the graph of f : (i) E = [a, b] (ii) E = {1}(iii) E = N (the set of natural numbers) and (iv) E = C (the Cantor set).

6. Show that f(x) = d(E, x) is continuous as a function of x, i.e. if xn → x asn →∞ then d(E, xn) → d(E, x) as n →∞.

11

MATHEMATICS 242

Eighth Assignment

Due 20 March 1992

1. If f is a continuous function on an interval [a, b] then the set

{x ∈ [a, b] : f(x) = k}

is closed.


{x ∈ [a, b] : f(x) ≥ k}

is closed.


{x ∈ [a, b] : f(x) > k}

is open.

4. If f is uniformly continuous on an open interval (a, b) then f is boundedthere.

5. If f is uniformly continuous on an open interval then lim f(xn) exists forevery sequence xn ∈ (a, b) with xn → a.

12

MATHEMATICS 242

Final Examination

8 April 1992

1. (a) Define what is meant by a Cauchy sequence.(b) Is your definition equivalent to the following statement for a sequence {xn}:for every ε > 0 and for every integer p there is an integer N so that

|xn+p − xn| < ε

for all n ≥ N?(c) Prove that every Cauchy sequence converges.

2. Define what is meant by the assertion that the series∑∞

n=1 an converges.Define what is meant by the assertion that the series

∑∞n=1 an converges abso-

lutely.

(i) Prove that if∑∞

n=1 an converges then limn→∞ an = 0 but not conversely.

(ii) Show that if {an} is a sequence of positive numbers with

an+1

an≤ 1

2

for all n then∑∞

n=1 an converges.

(iii) Suppose that {an} is a sequence of positive numbers with

an+1

an< 1

for all n. Can you conclude that∑∞

n=1 an converges?

13

3. Define what is meant by a point of accumulation of a set.Give an example of each of the following or else prove that such cannot exist:

(i) An infinite set with no point of accumulation.

(ii) An infinite set with countably many points of accumulation.

(iii) An infinite set with uncountably many points of accumulation.

(iv) An uncountable set with no point of accumulation.

4. Define what is meant by saying that f is a continuous function on an intervalI. Define what is meant by saying that f is a uniformly continuous function onan interval I.

(i) Give an example of a function which is continuous on (0, 1) but not uniformlycontinuous there. (Prove your statement.)

(ii) Suppose that f is continuous on the whole real line and that f(r) = r forevery rational number r. What can you conclude?

(ii) Suppose that f is continuous on the whole real line. Show that the set

{x : f(x) <12}

is open.

5. Let f : [0, 1] → [0, 1] be continuous.

(a) Show that f has a fixed point (i.e. a point x0 ∈ [0, 1] such that f(x0) = x0).

(b) Show that the set of all fixed points of f is closed.

(c) Suppose that the sequence {xn} defined by x1 = θ ∈ [0, 1], xn+1 = f(xn)converges. Show that limn→∞ xn is a fixed point of f .

14

MATHEMATICS 242

First Midterm Examination

February 17, 1992

1. Define each of the following terms: “sequence of real numbers”, “boundedsequence”, “limn→∞ xn = α”, and “Cauchy sequence”.

2. Prove directly from the definition that the sequence

xn =n + 12n

is convergent.

3. For what values of the starting point x1 = θ is the sequence

x1 = θ , xn+1 = (xn)−1n = 1, 2, 3, . . .

convergent?

4. Define what is meant by a “convergent series” and by an “absolutely con-vergent series”.

5. Let {an} be a sequence of positive numbers.(a) If limn→∞ n2an = 0 what (if anything) can be said about the series

∑∞n=1 an.

(b) If limn→∞ nan = 0 what (if anything) can be said about the series∑∞

n=1 an.(c) If we drop the assumption about the sequence {an} being positive doesanything change?

15

MATHEMATICS 242

Second Midterm Examination

9 March 1992

1. (a) Define what is meant by the expression

limt→t0

f(t) = x

for a function f .(b) Using your definition and not using any other theory prove that

limt→ 1

2

2t + 1 = 2.

2. (a) Define what is meant by a point of accumulation of a set.(b) If S is a bounded nonempty set and x = sup S show that either x ∈ S or elsex is a point of accumulation of S. Give examples to show that both possibilitiesmay occur.

3. Let G be an open set, F a closed set and {xn} a sequence of real numbersconverging to a number x. Prove (or disprove):

(i) If x ∈ G then there must exist an integer m so that xn ∈ G for all n ≥ m.

(ii) If x ∈ F then there must exist an integer m so that xn ∈ F for all n ≥ m.

(iii) If every xn 6∈ F then necessarily x 6∈ F .

(iv) If every xn ∈ G then necessarily x ∈ G.

16

MATHEMATICS 242

Final Examination

December 10, 1994


lim1sn

=1

lim sn

for a sequence {sn} under appropriate hypotheses. Include an example to showthat your hypotheses cannot be dropped.


limx→a

1f(x)

=1

limx→a f(x)

for a function f under appropriate hypotheses. Include an example to show thatyour hypotheses cannot be dropped.

3. Prove the limit comparison test for series:

Let {an} and {bn} be sequences of positive numbers for which the limit

limn→+∞

an

bn

exists. Show that∑∞

n=1 an converges if the series∑∞

n=1 bn converges.

4. Let an be a decreasing sequence of positive numbers. Is the following a validtest?

∑∞n=1 an is convergent if and only if limnan = 0.

(Is it a necessary condition? Is it a sufficient condition?)

17

5. Let f be a continuous function on an interval [a, b] and let

E = {x ∈ [a, b] : f(x) < 0}

(i) If c is a point of accumulation of the set E show that f(c) ≤ 0.

(ii) If d is a boundary point of the set E and d 6= a, b show that f(d) = 0.

(iii) Show that E can have no isolated points.

(iv) Is it possible that [a, b] \ E has isolated points?


|xn+2 − xn+1| ≤ α|xn+1 − xn|for all n.

(i) Show that every contractive sequence is convergent.

(ii) Give an example of a contractive sequence.

(iii) Give an example of a non-contractive sequence.

(iv) Is every convergent sequence contractive?

7. Let f be a continuous function on an interval [a, b] and let ε > 0 show thatthere are points

a = x0 < x1 < x2 < · · · < xn = b

so that the function g passing through the points (xi, f(xi)) (i = 0, 1, 2 . . . n)and linear on each interval [xi−1, xi] (n = 1, 2 . . . n) satisfies

|f(x)− g(x)| < ε

18

MATHEMATICS 242

First Assignment

Due September 16

Construct proofs for the following; in each case it is more the presentation andstyle of the proof that we wish to examine. Correct mathematical ideas howeverare not discouraged. 1

1. (Direct proof) For all real numbers x and y, x < y if and only if x3 < y3.



2 is irrational.



dn


1Some important information: homework (worth 10% of final grade) is mostly weekly anddue as stated. The first midterm (worth 20%) is Friday, October 7. The second midterm(worth 20%) is Wednesday, November 9. The final examination (worth 50%) is Saturday,December 10 at 8:30am.

You can contact the lecturer at 291-4233 or e–mail ([email protected]) for questions or anappointment. October 11 is the last day to drop. October 10 and November 11 are holidays.

19

MATHEMATICS 242

Second Assignment

Due September 23

1. Show that every nonempty set of integers that is bounded above has amaximal element.

2. If A is a nonempty bounded set of real numbers then −A = {x : −x ∈ A} isalso nonempty and bounded and

inf −A = − sup A.

3. If A and B are nonempty bounded sets of real numbers then

sup C = sup A + sup B

whereC = {x + y : x ∈ A and y ∈ B}.

4. The sequence {xn} defined by requiring x1 = 1 and xn+1 =√

2xn is boundedand monotonic.

20

MATHEMATICS 242

Third Assignment

Due September 30


















3. For any set S let S′ denote the set of its accumulation points. Give anexample that illustrates how each of the following can occur:

1. S′ = Ø.




21



7. S, S′ and S′′ are different.

22

MATHEMATICS 242

Fourth Assignment

Due October 14

1. Show that a finite union of compact sets is compact.

2. Show that an infinite union of compact sets need not be compact.

3. Give a nested sequence of bounded sets with empty intersection

4. Give a nested sequence of closed sets with empty intersection.

5. Let E be an uncountable set (bounded or not). Show that E has a point ofaccumulation.

6. Let E be a closed set. Show that E is compact if and only if every infinitesubset of E has at least one point of accumulation.

23

MATHEMATICS 242

Fifth Assignment

Due October 21









lim1sn

=1

lim sn

for a sequence {sn} under some appropriate hypotheses. (Include an exampleto show that your hypotheses cannot be dropped.)


xn+1 =x2

n − 2xn + 32

.


24

MATHEMATICS 242

Sixth Assignment

Due October 28








2. Prove (or disprove):






xn+1 =

√ab2 + xn

2

a + 1.


25

MATHEMATICS 242

Seventh Assignment

Due November 18

1. Which of these converge?

∞∑n=1

n(n + 1)(n + 2)2

,

∞∑n=1

3n(n + 1)(n + 2)n3√

n,

∞∑n=2

1ns log n

∞∑n=1

1.3. . . . (2n− 1)2.4. . . . 2n

√n

∞∑n=1

a1/n − 1,

∞∑n=2

1n(log n)t

∞∑n=2

1ns(log n)t

,

∞∑n=1

(1− 1

n

)n2


N∑n=1

1n(n + 1)(n + 2)

=14− 1

2(N + 1)(N + 2).

What is ∞∑n=1

1n(n + 1)(n + 2)

?



n is convergent.


a + ab + a2b + a2b2 + a3b2 + a3b3 + . . . .



x1 − 12(x1 + x2) +

13(x1 + x2 + x3)− 1

4(x1 + x2 + x3 + x4) . . .

converges.

26

MATHEMATICS 242

Eighth Assignment

Due November 25


{x ∈ [a, b] : f(x) = k}is closed.


{x ∈ [a, b] : f(x) ≥ k}is closed.


{x ∈ (a, b) : f(x) > k}is open.



6. If f is continuous on an interval [a, b] and ε > 0 then there is a continuouspiecewise linear function g such that

|f(x)− g(x)| < ε

for all x ∈ [a, b].

7. If f is continuous on [a, b] and f(a) < 0 < f(b) then there is a point c ∈ (a, b)with f(c) = 0. (Try to make the following argument into a correct proof: takethe largest point c in the set {x ∈ [a, b] : f(x) ≤ 0}. If f(c) < 0 then it can’t bethe largest point, so f(c) = 0.)

8. If f is continuous on [a, b] then it is bounded. (Try to make the followingargument into a correct proof: if it is not bounded then for each integer n thereis a point xn ∈ [a, b] with |f(xn)| > n. I can choose these points in such a waythat xn converges and then

limn→∞

f(xn) 6= f( limn→∞

xn)

which shouldn’t happen if f is continuous.)

27

MATHEMATICS 242

First Midterm

Friday October 7

1. If A and B are nonempty bounded sets of real numbers prove that



2. Define what is meant by an interior point of a set and by an open set.Give examples of each of the following or else prove that such a set cannot exist.(i) A nonempty bounded, open set that is denumerable.(ii) A nonempty bounded, closed set that is denumerable.(iii) A nonempty bounded, open set with no accumulation points.(iv) A nonempty bounded, closed set with no accumulation points.(v) Two sets A and B that are not open and yet A ∪B is open.(vi) Two sets A and B that are not open and yet A ∩B is open.

28

MATHEMATICS 242

Second Midterm

Wednesday November 9, 1994

1. A sequence of real numbers {sn} is said to be a null sequence if for everyε > 0 there is an integer m so that

|sn| < ε

whenever n ≥ m.Prove the following assertions directly from the definition (without using any

of the theory for convergent sequences).

1. If {sn} and {tn} are null sequences then so too is {sn + tn}.2. If {sn} and {tn} are null sequences then so too is {sntn}.3. If {sn} is a null sequence and 0 < tn < sn for every n ≥ n0 then {tn} is a

null sequence.

4. If {sn} is a null sequence then so too is {(sn)2}.

2. Define a sequence of real numbers recursively by writing x0 = 1 and

xn+1 =

√2 + xn

2

2.

Prove that the sequence {xn} is convergent and obtain its limit.

29

MATHEMATICS 242

Not an Assignment

November 1994

Each of these exercises relates some concept in the theory of real sets to oursequence theory.

1. Show that a set E is unbounded if and only if there is a sequence {xn} ofdistinct points of E with |xn| → +∞.

2. Show that x is a point of accumulation of a set E if and only if there is asequence {xn} of distinct points of E with xn → x.

3. Show that x is an interior point of a set E if and only if there is no sequence{xn} of points xn 6∈ E with xn → x.

4. Show that a sequence {xn} converges to a point x if and only if for everyopen set G containing x there is an integer N so that xn ∈ G for all n ≥ N .

5. Show that a set E is compact if and only if for every sequence {xn} of pointsof E there is a convergent subsequence with xnk

→ x where x ∈ E.

30

MATHEMATICS 242

First Assignment

Due January 18, 1996


1. (Direct proof) For all positive real numbers x and y, x < y if and only ifx2 < y2.



2 is irrational.



dn


2Some important information: homework (worth 10% of final grade) is mostly weekly anddue as stated. Two midterm examinations (20% each) and a final examination (50%).

You can contact the lecturer in 10,519 (M11:30, W11:30, F9:30) or call for appointment at291-4233 or use e–mail ([email protected]) for questions or an appointment.

31

MATHEMATICS 242

Second Assignment

Due January 26, 1996

Some important information: the first midterm is Friday, February 9 andthe second midterm is Friday, March 8.

1. Show that any interval (a, b) is equinumerous to the interval (0, 1). [Constructa specific bijection.]

2. Show that any two intervals (a1, b1) and (a2, b2) are equinumerous. [Do notconstruct a bijection; just use problem 1.]

3. Show that any interval (a, b) is equinumerous to the set of real numbers R.




6. Let C be a bounded set of real numbers. Show that

sup A = 2 sup C

andsupB = −2 inf C

whereA = {2x : x ∈ C} and B = {−2x : x ∈ C}.

32

MATHEMATICS 242

Third Assignment

Due February 2, 1996

Hand in answers to questions 2 and 3 only.


















3. For any set S let S′ denote the set of its accumulation points. Give anexample that illustrates how each of the following can occur:

1. S′ = Ø.



33




7. S, S′ and S′′ are different.

34

MATHEMATICS 242

Fourth Assignment

Due February 9









limsn

n= 0

for any bounded sequence {sn}.

35

MATHEMATICS 242

Fifth Assignment

Due February 16


xn+1 =x2

n − 2xn + 32

.

Discuss the convergence behaviour.(Hint: the cases θ < 1, 1 < θ < 3, etc. should be considered separately.)

36

MATHEMATICS 242

Sixth Assignment

Due February 23










xn+1 =

√ab2 + xn

2

a + 1.


37

MATHEMATICS 242

Seventh Assignment

Due March 1

1. Which of these converge?

∞∑n=1

n(n + 1)(n + 2)2

,

∞∑n=1

3n(n + 1)(n + 2)n3√

n,

∞∑n=2

1ns log n

∞∑n=1

1.3. . . . (2n− 1)2.4. . . . 2n

√n

∞∑n=1

a1/n − 1,

∞∑n=2

1n(log n)t

∞∑n=2

1ns(log n)t

,

∞∑n=1

(1− 1

n

)n2


N∑n=1

1n(n + 1)(n + 2)

=14− 1

2(N + 1)(N + 2).

What is ∞∑n=1

1n(n + 1)(n + 2)

?



n is convergent.


a + ab + a2b + a2b2 + a3b2 + a3b3 + . . . .



x1 − 12(x1 + x2) +

13(x1 + x2 + x3)− 1

4(x1 + x2 + x3 + x4) . . .

converges.

38

MATHEMATICS 242

Eighth Assignment

Due March 8

1. For what values of x do the following series converge?

∞∑n=2

xn

log n,

∞∑n=2

(log n)xn,

∞∑n=1

e−nx

2. Let an be a sequence of positive numbers and suppose that

limn→∞

nan = L

exists.What can you say about the convergence of the series

∑∞n=1 an if L = 0?

What can you say about the convergence of the series∑∞

n=1 an if L > 0?

3. Suppose that∞∑

n=1

an

is a convergent series of positive terms. Must

∞∑n=1

√an

also be convergent?

4. Prove the alternating series test directly from the Cauchy criterion.

39

MATHEMATICS 242

Ninth Assignment

Due March 22






{x ∈ (a, b) : f(x) > k}is open.



6. If f is continuous on an interval [a, b] and ε > 0 then there is a continuouspiecewise linear function g such that for all x ∈ [a, b]

|f(x)− g(x)| < ε




limn→∞


xn)


40

MATHEMATICS 242

Tenth Assignment

Due March 29

1. Suppose f : R → R is continuous. Prove or disprove:

1. f must be unbounded.

2. f cannot be uniformly continuous unless f is constant.

3. limn→∞ f( n√

n) = f(1).

4. f−1(E) is compact if E is compact.

5. f−1(E) is open if E is open.

6. f−1(E) is finite if E is finite.

7. f−1(E) is countable if E is countable.

8. f−1(E) is bounded if E is bounded.

2. Show that if f : [0, 1] → [0, 1] is a continuous function then f has a fixedpoint. Show that this is not necesarily true for discontinuous functions.

3. Let f : [0, 1] → [0, 1] be a continuous function and define g = f ◦ f . Showthat g must have a fixed point too and that every fixed point of f is also a fixedpoint of g but not conversely.

4. Show that if f and g are uniformly continuous on a set D then fg need notbe uniformly continuous on D.

4. Show that if f and g are uniformly continuous on an interval (a, b) then fgmust be uniformly continuous on (a, b).




limn→∞


xn)


41

MATHEMATICS 242


February 9

1. Let S be the set of all rational numbers q whose square q2 is smaller than 2.Prove that

supS =√

2.

2. Define what is meant by an interior point of a set E of real numbers. Definewhat is meant by an open set.

2. Define what is meant by an accumulation point of a set E of real numbers.Define what is meant by a closed set.

3. Prove that the union of any two open sets must be open.

4. Define what is meant by the limit of a sequence of real numbers.

5. Prove that if a sequence {sn} of real numbers converges to a positive numberL then there must exist an integer M so that sn > 0 for every n ≥ M .

6. Let E be a closed set and let {sn} be a sequence of real numbers convergingto a number L. Suppose that sn ∈ E for every n. Prove that L ∈ E.

42

MATHEMATICS 242


March 8

1. Prove that if a sequence of positive numbers sn converges to zero then sotoo does the sequence

√sn.


xn+1 =

√2 + xn

2

2.


3. Compute the sum of each of the following series:

∞∑n=2

(log(n + 1)− log(n))

∞∑n=2

(1

log(n)− 1

log(n + 1)

)

∞∑n=2

8(

19

)n


n=1 an and∑∞

n=1 bn are absolutely convergent. Show thatthen so too is the series

∑∞n=1 anbn.


n=1 an and∑∞

n=1 bn are both convergent. Show that it doesnot follow that the series

∑∞n=1 anbn is convergent.

43

MATHEMATICS 242

Suggested Review Problems

1. (a) Prove that a convergent sequence is bounded.

(b) Prove (directly from the definition) that a Cauchy sequence is bounded.

(c) Prove that a function uniformly continuous on an interval (a, b) is boundedthere.

2. (a) Prove thatlim

n→∞xnyn = ( lim

n→∞xn)( lim

n→∞yn)

if the sequences {xn} and {yn} are convergent.

(b) Prove thatlimx→c

f(x)g(x) = (limx→c

f(x))( limx→c

g(x))

if both of these limits exists.





44

4. (a) Prove that the series∞∑2

1n(lnn)p

converges for all p > 1 and diverges for all 0 < p ≤ 1.

(b) Let an be a sequence of postive numbers and write

Mn =log (1/nan)log(log n)

.

Show that if limn→∞Mn > 1 then∑∞

k=1 ak converges.Show that if Mn ≤ 1 for all sufficiently large n then

∑∞k=1 ak diverges.

5. Prove or disprove:Every bounded infinite set of real numbers has an accumulation point.Every infinite, countable set of real numbers has an accumulation point.Every uncountable set of real numbers has an accumulation point.Every nonempty open set of real numbers has an accumulation point.Every nonempty closed set of real numbers has an accumulation point.

6. Prove that if f is a continuous function on [a, b] and

∫ b

a

(f(x))2 dx = 0

then f is zero everywhere in [a, b].Is this true with the word “continuous” deleted?

45

MATHEMATICS 242

Final Examination

10 April 1996

1. Prove thatlim

n→∞anbn = ( lim

n→∞an)( lim

n→∞bn)

if the sequences {an} and {bn} are convergent.

2. Let the sequence {sn} be defined inductively by setting s1 = 0 and, for eachn ≥ 1,

sn+1 =(sn)2 + sn + 1

5.

Show the sequence converges and find its limit.

3. Suppose that limn→∞ xn = 0. Prove that

limn→∞

x1 + x2 + x3 · · ·+ xn

n= 0.

Is the converse also true?

4. Let xn be a sequence of positive numbers and write

zn =− log (nan)log(log n)

.

Show that if limn→∞ zn > 1 then∑∞

k=1 ak converges.

46

5. Let f and g be continuous functions on an interval [a, b]. Show that the set

{x ∈ [a, b] : f(x) = g(x)}

is closed and that the set

{x ∈ (a, b) : f(x) > g(x)}

is open.

6. Let E and D be sets of real numbers and suppose f : D → E is uniformlycontinuous on D. Show that for every Cauchy sequence {xn} in D the sequence{f(xn)} is a Cauchy sequence in E.

Does this statement remain true if f is continuous but not uniformly con-tinuous on D?

7. Let f be a continuous function defined on the interval [0, 1] and suppose that0 ≤ f(x) ≤ 1 for all x ∈ [0, 1] and that

∫ 1

0

f(x) dx =∫ 1

0

(f(x))2 dx.

What can you conclude?

47

MATHEMATICS 242

Final Examination

April 10 1988

Part I




xn+1 =

√2 + xn

2

2.


Part II

4. Define what is meant by each of the following terms: “ closed set of realnumbers” , “open set of real numbers”, “accumulation point”, “interior point”.

5. Prove or disprove the following statements:

(a) For a continuous function f on the real line the set of points

{x : f(x) = 1}is closed.

(b) For a continuous function f on the real line the set of points

{x : f(x) = 1}can have no points of accumulation.

(c) For a continuous function f on the real line the set of points

{x : f(x) = 1}can have no interior points.

(d) There is a closed set E that has infinitely many points but no points ofaccumulation.

48

(e) There is a closed set E that has countably many points of accumulation.

(f) There is an open set E that has no points of accumulation.

(g) There is an uncountable set E that has no points of accumulation.

Part III

6. Define what is meant by the uniform convergence of a sequence of functions.Give an example of a sequence of functions converging pointwise on the interval[1, 2] but not uniformly there.

7. Prove that a sequence {fn} of bounded functions that converges uniformlyon an interval [a, b] has the following property: there is a positive number M sothat

|fn(x)| < M

for every x ∈ [a, b] and for each n = 1, 2, 3, . . . .

49

MATHEMATICS 242

Old Final Examinations

8 April 19921. (a) Define what is meant by a Cauchy sequence.(b) Is your definition equivalent to the following statement for a sequence {xn}:for every ε > 0 and for every integer p there is an integer N so that

|xn+p − xn| < ε





lutely.




an+1

an≤ 1

2


n=1 an converges.


an+1

an< 1


n=1 an converges?

50










{x : f(x) <12}

is open.





April 10 1988Part I




xn+1 =

√2 + xn

2

2.


51

Part II




{x : f(x) = 1}

is closed.


{x : f(x) = 1}

can have no points of accumulation.


{x : f(x) = 1}

can have no interior points.





Part III



|fn(x)| < M


52

December 7 19871. Prove that a sequence of real numbers is convergent if and only if it is a

Cauchy sequence.


xn+1 =

√ab2 + xn

2

a + 1.




(a)∞∑

i=1

ai/i

(b)∞∑

i=1

ai2

(c)∞∑

i=1

√ai

(d)∞∑

i=1

ai

ai + 1



{x : 0 ≤ f(x) ≤ 1}

is closed.



53









54

MATHEMATICS 242

First Assignment

Due 19 January 1989

Construct proofs for the following; in each case it is more the presentation andstyle of the proof that we wish to examine. Correct mathematical ideas howeverare not discouraged.




2 is irrational.



dn


55

MATHEMATICS 242

Second Assignment

Due 26 January 1989







4. The sequence {xn} defined by requiring x1 = 1 and xn+1 =√

2xn is boundedand montonic.

56

MATHEMATICS 242

Third Assignment

Due 2 February 1989


lim1sn

=1

lim sn

for a sequence {sn} under the appropriate hypotheses. Include an example toshow that your hypotheses cannot be dropped.


xn+1 =x2

n − 2xn + 32

.




57

MATHEMATICS 242

Fourth Assignment


Only a fraction of these need be submitted for grading. The TA will indicatewhich ones that he wishes to see.

1. Which of these converge

∞∑n=1

n(n + 1)(n + 2)2

,

∞∑n=1

3n(n + 1)(n + 2)n3√

n,

∞∑n=2

1ns log n

∞∑n=1

1.3. . . . (2n− 1)2.4. . . . 2n

√n

∞∑n=1

a1/n − 1,

∞∑n=2

1n(log n)t

∞∑n=2

1ns(log n)t

,

∞∑n=1

(1− 1

n

)n2

2. If∑∞

n=1 an diverges and all an > 0 then what can be said about

∞∑n=1

an

1 + nan,

∞∑n=1

an

1 + n2an?


x1 − 12(x1 + x2) +

13(x1 + x2 + x3)− 1

4(x1 + x2 + x3 + x4) . . .

converges.

4. If∑∞

n=1 an converges and {bn} is monotonic and bounded then∑∞

n=1 anbn

converges (Abel, 1826).

58

MATHEMATICS 242

Fifth Assignment























59

3. Prove that the following definitions for “accumulation point” are equivalent:

1. A point x0 is an accumulation point of a set A provided that every deletedneighbourhood of x0 contains some point of A.

2. A point x0 is an accumulation point of a set A provided that every neigh-bourhood of x0 contains two points of A.

3. A point x0 is an accumulation point of a set A provided that every neigh-bourhood of x0 contains infinitely many points of A.

4. A point x0 is an accumulation point of a set A provided that there is asequence {xn} of distinct points of A such that limn→∞ xn = x0.


1. S′ = Ø.






60

MATHEMATICS 242

Sixth Assignment

Due 6 March 1989

1. Show that the set of accumulation points of any set is closed.

2. Let {xn} be a convergent sequence and let E be the range of the sequence.What is the closure of E?

3. Show that every uncountable set has a point of accumulation.

4. Show that every uncountable set has uncountably many accumulation points.

5. Show that the only sets of real numbers that are both open and closed are ∅and R itself.

6. Let E be a bounded, nonempty open set. Show that supE and inf E arepoints of accumulation of E neither of which belongs to E.

7. The distance between a point x and a closed E is defined to be

d(E, x) = inf{|x− y| : y ∈ E}.

Show that, for a closed set E, x ∈ E if and only if d(E, x) = 0. Is this true if Eis not closed?

8. Let x be a point that is not in a closed set E. Show that there is a pointe ∈ E such that d(E, x) = |x− e|. Is e unique?

9. Show that d(E, x) is continuous as a function of x, i.e. if xn → x as n →∞then d(E, xn) → d(E, x) as n →∞.

61

MATHEMATICS 242

Seventh Assignment

Due 27 March 1989







{x ∈ [a, b] : f(x) > k}is open.




|f(x)− g(x)| < ε




limn→∞


xn)


62

MATHEMATICS 242

Eighth Assignment

Due 3 April 1989


1. Let {fn} and {gn} be sequences of functions converging uniformly on a setE. Show that {fn + gn} also converges uniformly on E. Is the same true for{fngn} ?

2. Let {fn} be a sequence of nondecreasing functions on an interval [a, b]. Iflimn→∞ fn(x) = 0 for every point x ∈ [a, b] then {fn} converges uniformly on[a, b]. Is the same statement true if (i) the functions are not monotonic? (ii) ifthe interval [a, b] is replaced by (−∞, +∞)?

3. Let {fn} be a sequence of functions uniformly continuous on a set E anduniformly convergent on E to a function f . Show that f itself must be uniformlycontinuous on E as well.

4. Show that

limn→∞

n∑

k=0

xk =1

1− x

uniformly on [−δ,+δ] for every 0 < δ < 1 but not for δ = 1.

5. Let {fn} be a sequence of bounded functions on an interval [a, b]. Iflimn→∞ fn(x) = f(x) uniformly for x ∈ [a, b] then f must be bounded too.Is the same statement true if (i) the convergence is not uniform? (ii) if theinterval [a, b] is replaced by (−∞,+∞)?

63

6. Let {fn} be a sequence of continuous functions converging uniformly on(−∞, +∞) to a function f . Show that

limn→∞

fn

(x +

1n

)= f(x)

for each x.

64

MATHEMATICS 242

Assignment

29 October 1987

1. Prove that the following definitions for “accumulation point” are equiva-lent:

(a) A point x0 is an accumulation point of a set A provided that everydeleted neighbourhood of x0 contains some point of A.

(b) A point x0 is an accumulation point of a set A provided that everyneighbourhood of x0 contains two points of A.

(c) A point x0 is an accumulation point of a set A provided that everyneighbourhood of x0 contains infinitely many points of A.

(d) A point x0 is an accumulation point of a set A provided that there isa sequence {xn} of disinct points of A such that limn→∞ xn = x0.


(a) S′ = Ø.

(b) S′ contains just one point.

(c) S′ contains exactly two points.

(d) S′ is countably infinite.

(e) S′ is uncountable.

(f) S′ is nonempty but (S′)′= Ø.

65

MATHEMATICS 242

Review Problems

December 1 1987

1. [Prepared proof.] Prove that a sequence of real numbers is convergent ifand only if it is a Cauchy sequence.

2. Give definitions for each of the following terms: open set, closed set, com-pact set, point of accumulation, interior point, derived set.


(a) E has infinitely many points but no interior points.(b) E has infinitely many points but no points of accumulation.(c) E is open and unbounded.(d) E is closed and unbounded.(e) E has infinitely many points of accumulation but no interior points.(f) E is open but has no points of accumulation.(g) E is closed but has no points of accumulation.(h) E is compact and has no interior points.(i) E, E′ and E′′ are different.(j) E is countable and E′ = {0, 1}.(k) E is countable and E′ = [0, 1].(l) E is countable and E′ = (0, 1).

4. Prove that for a continuous function f on the real line the set of points

{x : 0 < f(x) < 1}is open.

5. Prove that for a continuous function f on the real line the set of points

{x : 0 ≤ f(x) ≤ 1}is closed.

6. Define uniform continuity and give an example of a function which iscontinuous but not uniformly continuous on the interval (−1, 1). Provethat your example is not uniformly continuous.

7. Prove directly from your definition of uniform continuity that the functionf(x) = x2 is uniformly continuous on (−1, 1).

8. Prove directly from your definition of uniform continuity that if a functionf is uniformly continuous on (−1, 1) then it must be bounded there.

66

MATHEMATICS 242

Final Examination

December 7 1987



xn+1 =

√ab2 + xn

2

a + 1.




(a)∞∑

i=1

ai/i

(b)∞∑

i=1

ai2

(c)∞∑

i=1

√ai

(d)∞∑

i=1

ai

ai + 1



{x : 0 ≤ f(x) ≤ 1}

is closed.


67










68

MATHEMATICS 242

First Midterm

8 February 1989

1. Directly from the definitions of the terms (i.e. without appealing to any ofthe general theory) prove (or disprove):

(i) Every convergent sequence is Cauchy.

(ii) Every Cauchy sequence is bounded.

(iii) Every bounded sequence is Cauchy.

(iv) If {sn} is Cauchy then so too is {|sn|}.

(iv) If {|sn|} is Cauchy then so too is {sn}.



xn+1 =

√ab2 + xn

2

a + 1.


69

MATHEMATICS 242

Second Midterm

15 March 1989

1. Let {an} and {bn} be sequences of positive numbers for which the limit

limn→+∞

an

bn

exists. Show that∑∞

n=1 an converges if the series∑∞

n=1 bn converges.

2. Show that the series ∞∑n=1

xn

1 + xn

diverges if xn → 0 but∑∞

n=1 xn diverges.




(iii) If every xn ∈ F then necessarily x ∈ F .


70

MATHEMATICS 242

Old Final Examinations

8 April 19921. (a) Define what is meant by a Cauchy sequence.(b) Is your definition equivalent to the following statement for a sequence {xn}:for every ε > 0 and for every integer p there is an integer N so that

|xn+p − xn| < ε





lutely.




an+1

an≤ 1

2


n=1 an converges.


an+1

an< 1


n=1 an converges?

71










{x : f(x) <12}

is open.





April 10 1988Part I




xn+1 =

√2 + xn

2

2.


72

Part II




{x : f(x) = 1}

is closed.


{x : f(x) = 1}

can have no points of accumulation.


{x : f(x) = 1}

can have no interior points.





Part III



|fn(x)| < M


73

December 7 19871. Prove that a sequence of real numbers is convergent if and only if it is a

Cauchy sequence.


xn+1 =

√ab2 + xn

2

a + 1.




(a)∞∑

i=1

ai/i

(b)∞∑

i=1

ai2

(c)∞∑

i=1

√ai

(d)∞∑

i=1

ai

ai + 1



{x : 0 ≤ f(x) ≤ 1}

is closed.



74









75

MATHEMATICS 242

Final Examination

December 10, 1998

Use two exam booklets. Answer questions 1–6 in Booklet #1 andquestions 7–11 in Booklet #2.

1. Give a precise definition for what is meant by the limit of a sequence of real [1 points]numbers.

2. Using your definition prove that if {sn} is a convergent sequence of positive [4 points]real numbers then the limit must be nonnegative.

3. Give a precise definition for what is meant by an open set. [1 points]

4. Using your definition prove that the intersection of any finite number of open [5 points]sets is again open.

5. Show that the intersection of an infinite number of open sets need not be [4 points]either open or closed.

6. A sequence is defined recursively by requiring x1 = 1 and xn+1 =√

2xn for [8 points]all n ≥ 1. Show that this sequence is convergent and find its limit.

76

Answer questions questions 7–11 in Booklet #2.

7. Let f : IR → IR. Show directly that the following two definitions of continuity [8 points]are equivalent:

(A) f is continuous at x0 if for every ε > 0 there is a δ > 0 so that

|f(x)− f(x0)| ≤ ε

whenever |x− x0| < δ.(B) f is continuous at x0 if for every sequence of real numbers {xn} conver-

gent to x0, limn→∞ f(xn) = f(x0).

8. Let f : E → IR, where E is a compact set, have this property: for every [8 points]x0 ∈ IR there is an integer N (which may depend on x0) such that

f(x) < N

if |x− x0| < 1/N and x ∈ E. Show that f must be bounded above on E.

Show that this conclusion need not be valid if E is closed but not bounded.

Show that this conclusion need not be valid if E is bounded but not closed.

9. Compute [4 points]lim

x→∞(f(x + a)− f(x− a))

if f is an everywhere differentiable function for which limx→∞

f ′(x) = L.

10. Suppose that f is a function that can be written in the form [4 points]

f(x) = x2(x2 − 1)p(x)

for some polynomial p(x). Show that there is a point ξ with |ξ| < 1 so thatf ′′(ξ) = 0.

11. A function f : IR → IR is said to have an isolated zero at a point x0 if [4 points]f(x0) = 0 but x0 is an isolated point in the set {x : f(x) = 0}. Show that afunction can have at most countably many isolated zeros.

[Total con-sidered as 50points]

77

MATHEMATICS 242


October 5, 1998

1. Give a precise definition for what is meant by the convergence of a sequence{sn} to a number L.

2. Give a precise definition for what is meant by the divergence of a sequence{sn} to ∞.

3. Using your definition and no other facts from the theory prove the followingresult:

If limn→∞ sn = 1 then limn→∞(sn)2 = 1.

4. Using your definition and no other facts from the theory prove the followingresult:

If {sn} is a sequence of positive numbers and limn→∞ sn = 0 thenlimn→∞ sn = ∞.

5. Define a sequence {sn} recursively by setting s1 = 1 and

sn =√

1 + sn−1.

Show that this sequence converges and determine its limit.

6. [time permitting] Can you handle question #5 if, instead, s1 = α whereα ≥ −1?

78

MATHEMATICS 242


October 30, 1998

1. Give a precise definition for what is meant by a point of accumulation of aset.

2. Let x be a point that does not belong to the set E. Show that x is a pointof accumulation of E if and only if x is not an interior point of IR \ E.

3. Let E be a nonempty set of real numbers and define the function

f(x) = inf{|x− e| : e ∈ E}.

(a) Show that f(x) = 0 for all x ∈ E.(b) Show that f(x) = 0 if and only if x ∈ E.(c) Show for any closed set E that

{x ∈ IR : f(x) > 0} = (IR \ E).

4. Let f : E → IR have this property. For every e ∈ E there is an ε > 0 so that

f(x) > ε if x ∈ E ∩ (e− ε, e + ε).

Show that if the set E is compact then there is some positive number c so that

f(e) > c

for all e ∈ E.Show that if E is not compact this conclusion may not be valid.

79

MATHEMATICS 242

Final Assignment

Due December 3

1. Suppose that f : IR → IR and both f ′ and f ′′ exist everywhere. Show thatif f has three zeros then there must be some point ξ so that f ′′(ξ) = 0.

2. A function f is said to satisfy a Lipschitz condition on an interval [a, b]if |f(x) − f(y)| ≤ M |x − y| for all x, y in the interval. Show that if f isassumed to be continuous on [a, b] and differentiable on (a, b) then this conditionis equivalent to the derivative f ′ being bounded on (a, b).

Give an example of a continuous function on an interval [a, b] that is notLipschitz.

3. Suppose that f is a continuous function such that limx→x0 f ′(x) does exist.Does it follow that f ′(x0) also exists? Does it follow that f ′ is continuous atx0?

4. Suppose f satisfies the hypotheses of the mean value theorem on [a, b] . LetS be the set of all slopes of chords determined by pairs of points on the graphof f and let D be the set {f ′(x) : x ∈ (a, b)}.

1. Prove S ⊂ D.

2. Give an example to show that D can contain numbers not in S.

5. Suppose that f is differentiable on [0,∞) and that

limx→∞

f ′(x) = C.

Determinelim

x→∞[f(x + a)− f(x)].

6. Suppose that f , g : IR → IR and both have continuous derivatives and thedeterminant

φ(x) =∣∣∣∣

f(x) g(x))f ′(x) g′(x)

∣∣∣∣is never zero. Show that between any two zeros of f there must be a zero of g.

7. Consider the functionF (x) =

3x − 2x

x

defined except at x = 0.

80

1. What value should be assigned to F (0) so that the resulting function iscontinuous?

2. Is this function now also differentiable at x = 0 and if so what is itsderivative?

3. Would it be correct as an answer to part (b) merely to compute F ′(x) bythe rules of the calculus and then determine limx→0 F ′(x)?

81

MATHEMATICS 242

Second Assignment

Due 24 September


2. Show for every finite, nonempty set of real numbers A that

sup A = max A.

Give an inductive proof on the number of elements of A, i.e., for every n ∈ Nshow that every set with n elements has this property.






5. Show that the sequence {xn} defined by requiring x1 = 1 and xn+1 =√

2xn

is bounded and monotonic.

82

MATHEMATICS 242

Third Assignment

Due October 2

1. Define

e = limn→∞

(1 +

1n

)n

.

Show that 2 < e < 3. Show that

limn→∞

(1 +

12n

)n

=√

e.

Check the simple identity(

1 +2n

)=

(1 +

1n + 1

)(1 +

1n

)

and use it to show that

limn→∞

(1 +

2n

)n

= e2.


xn+1 =x2

n − 2xn + 32

.


3. For any convergent sequence {an} write sn = (a1 + a2 + . . . an)/n, thesequence of averages. Show that

limn→∞

an = limn→∞

sn.

Give an example to show that {sn} could converge even if {an} diverges.Show that every nonempty set of integers that is bounded above has a max-

imal element.


sup A = max A.


83







2xn


84

MATHEMATICS 242

Fourth Assignment

Due October 7

1. Is the definition of a Cauchy sequence stronger, weaker or equivalent to:

for every ε > 0 and for every integer p there is an integer N so that

|xn+p − xn| < ε

for all n ≥ N?

2. Prove (or disprove) directly from the definition of a Cauchy sequence (i.e.,without using any theory):




3. If x2k → L and x2k+1 → L as k →∞ then xk → L.Show that every nonempty set of integers that is bounded above has a max-

imal element.


sup A = max A.








2xn


85

MATHEMATICS 242

Fifth Assignment

Due October 16









(iii) If every xn ∈ F then necessarily x ∈ F .


86

MATHEMATICS 242

Sixth Assignment

Due October 23

1. Give an example of a sequence of open sets G1, G2, G3, . . . whose intersectionis neither open nor closed.

Give an example of a sequence of closed sets F1, F2, F3, . . . whose union isneither open nor closed.

2. Let A and B be nonempty sets of real numbers and let

δ(A,B) = inf{|a− b| : a ∈ A, b ∈ B}.

δ(A,B) is often called the “distance” between the sets A and B.

1. Prove δ(A,B) = 0 if A ∩B 6= ∅.2. Give an example of two closed, disjoint sets in IR for which δ(A,B) = 0.

3. Prove that if A is compact, B is closed and A ∩B = ∅, then δ(A,B) > 0.

3. Give an example of an open covering of the set Q of rational numbers thatdoes not reduce to a finite subcover.

4. We have seen in the text that the following four conditions on a set A ⊂ IRare equivalent:

(i) A is closed and bounded

(ii) Every infinite subset of A has a limit point in A.

(iii) Every sequence of points from A has a subsequence converging to a pointin A

(iv) Every open cover of A has a finite subcover.

Prove directly that (ii) implies (iii), (ii) implies (iv) and (iii) implies (iv).

87

MATHEMATICS 242

Seventh Assignment

Due November 6

1. Prove the validity of the limit

limx→x0

(ax + b) = ax0 + b.

2. Prove the existence of the limit

limx→−4

x2.

3. Suppose that x0 is a point of accumulation of both A and B and thatf : A → IR and g : B → IR. We insist that f and g must agree in the sense thatf(x) = g(x) if x is in both A and B.

1. What conditions on A and B ensure that if limx→x0 f(x) exists so toomust limx→x0 g(x)?

2. What conditions on A and B ensure that if limx→x0 f(x) and limx→x0 g(x)both exist then they must be equal.

4. Give an example of a limit for which

limx→x0

F (f(x)) 6= F

(lim

x→x0f(x)

)

even though both of the limits in the statement do exist.

88

MATHEMATICS 242

Eighth Assignment

Due November 13

1. A function f is said to be symmetrically continuous at a point x if

limh→0

f(x + h)− f(x− h) = 0.

Show that if f is continuous at a point then it must be symmetrically continuousthere and that the converse does not hold.

2. Let f be defined on the set containing the points 0, ±1, ±1/2, ±1/4, ±1/8,. . .±1/2n, . . . only. What values can you assign at these points that will makethis function continuous everywhere where it is defined?

3. Let x0 ∈ IR. Below are four δ − ε conditions on a function f : IR → IR.

1. There exists ε > 0 such that for each δ > 0, if |x − x0| < δ then|f(x)− f(x0)| < ε.

2. There exists ε > 0 such that for each δ > 0, if |f(x) − f(x0)| < δ then|x− x0| < ε.

3. There exists ε > 0 such that for each δ > 0, if |x − x0| < ε then|f(x)− f(x0)| < δ.

4. There exists ε > 0 such that for each δ > 0, if |f(x) − f(x0)| < ε then|x− x0| < δ.

Which, if any, of these conditions imply continuity of f at x0? Which, if any,are implied by continuity at x0?

4. Suppose f is continuous and one-to-one on IR. Prove that the inverse functionf−1 is continuous on f(IR) (this is the range of f).

89

MATHEMATICS 242

Ninth Assignment Revised

Due November 20

1. Suppose f is uniformly continuous on each of the compact sets X1, X2, . . . ,Xn. Prove f is uniformly continuous on the set X =

⋃ni=1 Xi.

Show that this need not be the case if the sets Xk are not closed and neednot be the case if the sets Xk are not bounded.

2. Give an example of a function f that is continuous on IR and a sequence ofcompact intervals X1, X2, . . . , Xn, . . . on each of which f is uniformly contin-uous, but for which f is not uniformly continuous on X =

⋃∞i=1 Xi.

3. Prove that a function that is uniformly continuous on a bounded set E isbounded. (Is this true without the first “bounded”?)

4. Let f : IR → IR be a continuous nonnegative function with the property that

limx→∞

f(x) = limx→−∞

f(x) = 0.

Show that f has an absolute maximum. Need it have also an absolute minimum?

90

MATHEMATICS 242

December 7 1987



xn+1 =

√ab2 + xn

2

a + 1.




(a)∞∑

i=1

ai/i

(b)∞∑

i=1

ai2

(c)∞∑

i=1

√ai

(d)∞∑

i=1

ai

ai + 1



{x : 0 ≤ f(x) ≤ 1}

is closed.


91










92

MATHEMATICS 242

Final Examination

April 10 1988

Part I




xn+1 =

√2 + xn

2

2.


Part II




{x : f(x) = 1}is closed.


{x : f(x) = 1}can have no points of accumulation.


{x : f(x) = 1}can have no interior points.


93




Part III



|fn(x)| < M


94

MATHEMATICS 242

Final Examination

8 April 1992

1. (a) Define what is meant by a Cauchy sequence.(b) Is your definition equivalent to the following statement for a sequence {xn}:for every ε > 0 and for every integer p there is an integer N so that

|xn+p − xn| < ε





lutely.




an+1

an≤ 1

2


n=1 an converges.


an+1

an< 1


n=1 an converges?

95










{x : f(x) <12}

is open.





96

MATHEMATICS 242

First Assignment

Due September 16





2 is irrational.



dn


3Some important information: homework (worth 10% of final grade) is mostly weekly anddue as stated. The first midterm (worth 20%) is Friday, October 7. The second midterm(worth 20%) is Wednesday, November 9. The final examination (worth 50%) is Saturday,December 10 at 8:30am.

You can contact the lecturer at 291-4233 or e–mail ([email protected]) for questions or anappointment. October 11 is the last day to drop. October 10 and November 11 are holidays.

97

MATHEMATICS 242

Final Examination

10 April 1996

1. Prove thatlim

n→∞anbn = ( lim

n→∞an)( lim

n→∞bn)

if the sequences {an} and {bn} are convergent.

2. Let the sequence {sn} be defined inductively by setting s1 = 0 and, for eachn ≥ 1,

sn+1 =(sn)2 + sn + 1

5.

Show the sequence converges and find its limit.

3. Suppose that limn→∞ xn = 0. Prove that

limn→∞

x1 + x2 + x3 · · ·+ xn

n= 0.

Is the converse also true?

4. Let xn be a sequence of positive numbers and write

zn =− log (nan)log(log n)

.

Show that if limn→∞ zn > 1 then∑∞

k=1 ak converges.

98

5. Let f and g be continuous functions on an interval [a, b]. Show that the set

{x ∈ [a, b] : f(x) = g(x)}

is closed and that the set

{x ∈ (a, b) : f(x) > g(x)}

is open.

6. Let E and D be sets of real numbers and suppose f : D → E is uniformlycontinuous on D. Show that for every Cauchy sequence {xn} in D the sequence{f(xn)} is a Cauchy sequence in E.

Does this statement remain true if f is continuous but not uniformly con-tinuous on D?


∫ 1

0

f(x) dx =∫ 1

0

(f(x))2 dx.


99

MATHEMATICS 242

Final Examination

December 10, 1998


1. Give a precise definition for what is meant by the limit of a sequence of real [1 points]numbers.

2. Using your definition prove that if {sn} is a convergent sequence of positive [4 points]real numbers then the limit must be nonnegative.

3. Give a precise definition for what is meant by an open set. [1 points]

4. Using your definition prove that the intersection of any finite number of open [5 points]sets is again open.

5. Show that the intersection of an infinite number of open sets need not be [4 points]either open or closed.

6. A sequence is defined recursively by requiring x1 = 1 and xn+1 =√

2xn for [8 points]all n ≥ 1. Show that this sequence is convergent and find its limit.

100


7. Let f : IR → IR. Show directly that the following two definitions of continuity [8 points]are equivalent:

(A) f is continuous at x0 if for every ε > 0 there is a δ > 0 so that

|f(x)− f(x0)| ≤ ε

whenever |x− x0| < δ.(B) f is continuous at x0 if for every sequence of real numbers {xn} conver-

gent to x0, limn→∞ f(xn) = f(x0).

8. Let f : E → IR, where E is a compact set, have this property: for every [8 points]x0 ∈ IR there is an integer N (which may depend on x0) such that

f(x) < N

if |x− x0| < 1/N and x ∈ E. Show that f must be bounded above on E.

Show that this conclusion need not be valid if E is closed but not bounded.

Show that this conclusion need not be valid if E is bounded but not closed.

9. Compute [4 points]lim

x→∞(f(x + a)− f(x− a))

if f is an everywhere differentiable function for which limx→∞

f ′(x) = L.

10. Suppose that f is a function that can be written in the form [4 points]

f(x) = x2(x2 − 1)p(x)

for some polynomial p(x). Show that there is a point ξ with |ξ| < 1 so thatf ′′(ξ) = 0.

11. A function f : IR → IR is said to have an isolated zero at a point x0 if [4 points]f(x0) = 0 but x0 is an isolated point in the set {x : f(x) = 0}. Show that afunction can have at most countably many isolated zeros.

[Total con-sidered as 50points]

101

MATHEMATICS 242

Final Examination

April 26, 2000


1. Give a precise definition for what is meant by a Cauchy sequence of real [1 point]numbers.

2. Prove that {sn} is a convergent sequence of real numbers if and only if it is [10 points]a Cauchy sequence.

3. Give precise definitions for what is meant by an open set and by a closed set. [1 point]

4. Prove that a set of real numbers is open if and only if the complement of [10 points]that set is closed.

5. Give a definition for what it means for a function to be uniformly continuous [1 point]on an interval I.

6. Prove that a function f that is uniformly continuous on an interval (a, b) [8 points]must have the property that if a < xn < b and xn → a then necessarily thesequence {f(xn} is convergent.

7. Does question #6 also hold if f is continuous but not uniformly continuous [2 points]on (a, b)?

102


8. A sequence is defined recursively by requiring x1 = α > 0 and [10 points]

xn+1 =

√x2

n + β

1 + β2

for all n ≥ 1. Determine whether this sequence is convergent and, if so, find itslimit.

9. Prove that every bounded infinite set of real numbers must have a point of [8 points]accumulation. Suppose that a function f : IR → IR has this property: for everys ∈ IR there is an integer m so that f(x) < m if s−m−1 < x < s + m−1. Showthat f must be bounded above on every bounded interval.

10. Must the function in #9 be bounded above on (−∞,∞)? [5 points]

[“Total” 50points]

103

MATHEMATICS 242

Final Examination

April 19, 2000







6. Prove that a function f that is uniformly continuous on an interval (a, b) [8 points]must be bounded.

7. Give an example of a function f that is continuous on the interval (−1, 1) [2 points]but is not uniformly continuous.

104


8. A sequence is defined recursively by requiring x1 = β > 0 and [10 points]

xn+1 =

√x2

n + β

1 + β2


9. Prove that every bounded infinite set of real numbers must have a point of [8 points]accumulation.

10. Prove that every uncountable set of real numbers (bounded or not) must [5 points]contain a point that is a point of accumulation.


105

MATHEMATICS 242

First Assignment

Due in tutorial January 20





2 is irrational.



dn


106

MATHEMATICS 242

Second Assignment

Due in Tutorial January 27


sup A = max A.








2xn


107

MATHEMATICS 242

Third Assignment

Due in Tutorial February 3

1. Define

e = limn→∞

(1 +

1n

)n

assuming that this sequence does converge. Show that 2 < e < 3. Show that

limn→∞

(1 +

12n

)n

=√

e.


1 +2n

)=

(1 +

1n + 1

)(1 +

1n

)


limn→∞

(1 +

2n

)n

= e2.

2. A sequence {xn} is defined by starting at some value x1 = β and continuinginductively

xn+1 =x2

n − 2xn + 32

.

Discuss the convergence behaviour. (Hint: the cases β < 1, 1 < β < 3, etc.should be considered separately.)


limn→∞

an = limn→∞

sn.


imal element.


sup A = max A.


108







2xn


109

MATHEMATICS 242

Fourth Assignment


1. Show that every bounded monotonic sequence is Cauchy without using themonotone convergence theorem.

2. Show that every Cauchy sequence is bounded without using the fact thatconvergent sequences must be bounded.



Show that every nonempty set of integers that is bounded above has a max-imal element.


sup A = max A.








2xn


110

MATHEMATICS 242

Fifth Assignment






















13. E′ is countably infinite.

14. E′ is uncountable.

15. E′ is nonempty but (E′)′= Ø.

111

MATHEMATICS 242

Sixth Assignment

Due in Tutorial March 2


f(x) = inf{|x− e| : e ∈ E}.


{x ∈ IR : f(x) > 0} = (IR \ E).


f(x) > ε if x ∈ E ∩ (e− ε, e + ε).


f(e) > c



δ(A,B) = inf{|a− b| : a ∈ A, b ∈ B}.




4. Give an example of an open covering of the set CQ of rational numbers thatdoes not reduce to a finite subcover.

112

MATHEMATICS 242

Seventh Assignment


1. Prove the intermediate value theorem for continuous functions by using thefollowing “last point” argument: suppose that f(a) < 0 < f(b) and let z be thelast point in [a, b] where f(z) ≤ 0, i.e., let

z = sup{x ∈ [a, b] : f(x) ≤ 0}.

Show that f(z) = 0.

2. Show that the function f(x) = cos x defined for all x ∈ IR has a fixed pointα by applying the intermediate value property. Is α unique?

3. Let {xn} be defined by setting x1 = 1 and xn+1 = cos xn for all n.

(a) Show that if this is a convergent sequence then xn → α [same α as in #2].

(b) Show that there is a number 0 < β < 1 so that |xn−α| < βn. [Hint: use themean value theorem applied to the function f(x) = cos x on the interval [0, 1]and don’t forget that f(α) = α.]

(c) Now prove “again” that xn → α.

(d) For what value of N can you be sure that |xN − α| < 10−10?

(d) What is the best estimate for α that your calculator can give? How manytimes did you press the “cos” key?

113

MATHEMATICS 242

Eighth Assignment



{x ∈ [a, b] : f(x) = k}

is closed.


{x ∈ (a, b) : f(x) > k}

is open.


4. A function f : (a, b) → IR is said to be Lipschitz if there is a positive numberM so that |f(x)− f(y)| ≤ M |x− y| for all x, y ∈ (a, b). Show that a Lipschitzfunction must be uniformly continuous. Is the converse true?

5. Let f be a uniformly continuous function on a set E. Show that if {xn} is aCauchy sequence in E then {f(xn)} is a Cauchy sequence in f(E). Show thatthis need not be true if f is continuous but not uniformly continuous.

2.

3. (a) (b)

(d)

(d)

114

MATHEMATICS 242

Ninth Assignment

Due April 1, 20004

1. If f is continuous and m ≤ f(x) ≤ M for all x in [a, b] show that

m(b− a) ≤∫ b

a

f(x) dx ≤ M(b− a).

2. Calculate∫ 1

0xp dx (for whatever values of p you can manage) by partitioning

[0, 1] into subintervals of equal length.

3. Calculate∫ b

axp dx (for whatever values of p you can manage) by partitioning

[a, b] into subintervals [a, aq], [aq, aq2], . . . [aqn−1, b] where aqn = b.

4. (Mean Value Theorem for Integrals) If f is continuous on [a, b] showthat there is a point ξ in (a, b) so that

∫ b

a

f(x) dx = f(ξ)(b− a).

5. (Cauchy-Schwarz inequality) If f and g are continuous on an interval[a, b] show that

(∫ b

a

f(x)g(x) dx

)2

≤(∫ b

a

[f(x)]2 dx

)(∫ b

a

[g(x)]2 dx

)

6. Formulate a definition of the integral∫∞−∞ f(x) dx for a function continuous

on (−∞,∞). Supply examples of convergent and divergent integrals of thistype.

7. (Cauchy Criterion for Convergence) Let f : [a,∞) → IR be a continuousfunction. Show that the integral

∫∞a

f(x) dx converges if and only if for everyε > 0 there is a number M so that, for all M < c < d,

∣∣∣∣∣∫ d

c

f(x) dx

∣∣∣∣∣ < ε.

4Just kidding, make it the last week of classes.

115

MATHEMATICS 242

Seventh Assignment

Due 27 March 1989







{x ∈ [a, b] : f(x) > k}is open.




|f(x)− g(x)| < ε




limn→∞


xn)


116

MATHEMATICS 242


February 4, 2000

1. (a) Define what is meant by the expression

limn→∞

an = x

for a sequence of real numbers {an} and a real number x.(b) Prove that if limn→∞ an = 0 and {bn} is a bounded sequence then theproduct sequence {anbn} is convergent.(c) Is it true that if limn→∞ an = 1 and {bn} is a bounded sequence then theproduct sequence {anbn} is convergent?

2. For any two convergent sequences {an} and {bn} give a complete proof thatthe product sequence {anbn} is convergent.

[Hint: you may assume the theorem that asserts that every convergent se-quence is bounded.]

3. A sequence {xn} is defined by starting at x1 = 1 and continuing inductively

xn+1 =√

3 + xn.

(a) Prove inductively that {xn} is bounded.(b) Prove inductively that {xn} is increasing.(c) Show that {xn} is convergent and determine the value of the limit.



sup A = max A.






117



2xn


118

MATHEMATICS 242


March 10, 2000

1. (a) Define what is meant by a Cauchy sequence.

(b) Let {xn} be a sequence of real numbers with the property that

|xn+1 − xn| ≤ 12n

for all n = 1, 2, 3, . . . . Show that {xn} converges.

(c) Show that, for the sequence in (c),

x1 − 1 ≤ limn→∞

xn ≤ x1 + 1.

2. (a) Define what it means for a set G to be open.

(b) Prove or disprove that if G1, G2, G3, . . . are open sets then⋂m

k=1 Gk is openfor all positive integers m.

(c) Prove or disprove that if G1, G2, G3, . . . are open sets then⋂∞

k=1 Gk is open.

(d) Suppose that G is open and xn is a sequence converging to a point in G.Prove that there must be an integer m so that xk ∈ G for all k ≥ m.

3. Let f be a function defined on an interval [a, b] with the property thatfor all x ∈ [a, b] there is an integer p so that f(t) < p for all t ∈ [a, b] withx− 1/p < t < x + 1/p. Show that there is an integer P so that f(y) ≤ P for ally ∈ [a, b].



sup A = max A.




119





2xn


120

MATHEMATICS 242

Second Midterm

Wednesday November 9, 1994

1. A sequence of real numbers {sn} is said to be a null sequence if for everyε > 0 there is an integer m so that

|sn| < ε

whenever n ≥ m.Prove the following assertions directly from the definition (without using any

of the theory for convergent sequences).

1. If {sn} and {tn} are null sequences then so too is {sn + tn}.2. If {sn} and {tn} are null sequences then so too is {sntn}.3. If {sn} is a null sequence and 0 < tn < sn for every n ≥ n0 then {tn} is a

null sequence.

4. If {sn} is a null sequence then so too is {(sn)2}.


xn+1 =

√2 + xn

2

2.


121

MATHEMATICS 242


March 8

1. Prove that if a sequence of positive numbers sn converges to zero then sotoo does the sequence

√sn.


xn+1 =

√2 + xn

2

2.


3. Compute the sum of each of the following series:

∞∑n=2

(log(n + 1)− log(n))

∞∑n=2

(1

log(n)− 1

log(n + 1)

)

∞∑n=2

8(

19

)n


n=1 an and∑∞

n=1 bn are absolutely convergent. Show thatthen so too is the series

∑∞n=1 anbn.


n=1 an and∑∞

n=1 bn are both convergent. Show that it doesnot follow that the series

∑∞n=1 anbn is convergent.

122

MATHEMATICS 242

Final Examination

April 19, 2000

Answer all the True and False questions in Part A on the examination sheetitself. Do not show any justifications.

PART A. Each of these questions is either true or false. Circle the correctanswer. If it is true this means that you should be able to prove it; if it isfalse then you should have a counterexample in mind. You get one point forindicating correctly and you lose one point for an incorrect indication. Pleaseleave it blank if you are merely guessing or you may guess yourself to a negativegrade on this exam.

1. True or False? For any set E ⊂ IR if E is not closed then IR \ E is notopen.

2. True or False? For any open, bounded set E ⊂ IR the set E is compact.

3. True or False ? If E is closed then no point of E can be an interior point.

4. True or False ? If E is open then every point of E is an accumulationpoint.

5. True or False ? If E is not closed then some point of E must be aninterior point.

6. True or False ? If E is not open then some point of E must not be apoint of accumulation.

7. True or False ? If E is not compact then it must contain a sequence {xn}diverging to either +∞ or −∞.

8. True or False? For any set E ⊂ IR the set E′ is closed.

9. True or False? For any compact set E ⊂ IR the set E′ is compact.

10. True or False? For any open set G the set G \G′ is open.

11. True or False? For any open set G the set G′ \G is closed.

12. True or False? If E′ = ∅ then E must be closed.

13. True or False ? If E 6= E′ then E cannot be closed.

14. True or False ? If E ⊂ E′ then E must be closed.

15. True or False? If E′ ⊂ E then E must be closed.

123

16. True or False? If E1 and E2 are open, dense sets then so too is the setE1 ∪ E2.

17. True or False? If E1 and E2 are open, dense sets then so too is the setE1 ∩ E2.

18. True or False? If E1, E2, . . . , Ep is a finite sequence of compact setsthen the set

⋃pi=1 Ei is also compact.

19. True or False? If E1, E2, . . . , Ep is a finite sequence of compact setsthen the set

⋂pi=1 Ei is also compact.

20. True or False ? If E1, E2, . . . , is a sequence of compact sets then the set⋃∞i=1 Ei is also compact.

21. True or False? If E1, E2, . . . is a sequence of compact sets then the set⋂pi=1 Ei is also compact.

22. True or False? If A and B are open subsets of IR then the set IR\(A∪B)is closed.

23. True or False? If A and B are open subsets of IR then the set IR\(A∩B)is closed.

24. True or False? If A is an open set and B is a closed set then A \B mustbe open and B \A must be closed.

25. True or False ? If x0 is an isolated point of a set A and A ⊂ B then x0

must also be an isolated point of B.

26. True or False? If x0 is an accumulation point of a set A and A ⊂ B thenx0 must also be an accumulation point of B.

27. True or False? If x0 is an interior point of a set A and A ⊂ B then x0

must also be an interior point of B.

28. True or False? If f : [a, b] → IR is a continuous function then there mustbe a point c ∈ [a, b] with the value f(c) = (f(a) + f(b))/2.

29. True or False? If f : [a, b] → IR is a continuous function then

−∞ < inf{f(t) : a < t < b} < +∞.

30. True or False ? If f : [a, b] → IR is a continuous function then

< sup{f(t) : a < t < b} = f(b).

31. True or False ? If f : IR → IR is a continuous function then there is at > 0 so that |f(x)− f(y)| < 1 whenever |x− y| < t, x, y ∈ IR.

124

32. True or False? If f : [a, b] → IR is a continuous function then there is at > 0 so that |f(x)− f(y)| < 1 whenever |x− y| < t, x, y ∈ [a, b].

33. True or False? If f : IR → IR is continuous at a point x0 then for alla > 0 there is a b > 0 so that |f(x)− f(x0)| < a if |x− x0| < b.

34. True or False ? If f : IR → IR is continuous at a point x0 then for allc > 0 there is a d > 0 so that |f(x)− f(x0)| < d if |x− x0| < c.

35. True or False? If f : IR → IR is continuous and {xn} is a Cauchysequence then the sequence {f(xn)} must be convergent.

36. True or False ? If f : IR → IR is continuous and {xn} is a divergentsequence then the sequence {f(xn)} must be also be divergent.

37. True or False? If limk→∞ xk = δ then for all a > 0 there is an integer pso that |xi − δ| < a whenever i > p.

38. True or False? If limk→∞ xk = δ then for all a > 0 there is an integer pso that |xi − δ| ≤ a whenever i ≥ p.

39. True or False ? If limk →∞xk = δ then for all a > 0 there is an integerN so that |xk − a| < δ whenever k ≥ N .

40. True or False ? If the sequence {an} is convergent then so too must bethe sequence

bn =1

1 + an.

41. True or False ? If the sequence {an} is divergent then so too must be thesequence

bn =1

1 + an.

42. True or False? If the sequence {an} is convergent then so too must bethe sequence

bn =1

1 + an + (an)2.

43. True or False ? If the sequence bn = 1 + an + (an)2 is convergent then sotoo must be the sequence {an}.

44. True or False ? Every sequence has a Cauchy subsequence.

45. True or False ? Every bounded sequence has divergent subsequence.

46. True or False? Every divergent bounded sequence has a convergent sub-sequence.

47. True or False? Every unbounded sequence has a monotonic subsequence.

125

PART B. Give complete proofs for each of the following statements if it istrue or give a counterexample if you believe it is false.

1. A sequence of real numbers is convergent if and only if it is a Cauchysequence.

2. A function f : (a, b) → IR that is continuous must be uniformly continuouson (a, b).

3. A function f : (a, b) → IR that is uniformly continuous must be boundedon (a, b).






6. Prove that a function f that is uniformly continuous on an interval (a, b) [8 points]must be bounded.

7. Give an example of a function f that is continuous on the interval (−1, 1) [2 points]but is not uniformly continuous.

126

PART C.Solve the following problems.

8. A sequence is defined recursively by requiring x1 = β > 0 and [10 points]

xn+1 =

√x2

n + β

1 + β2


9. Prove that every bounded infinite set of real numbers must have a point of [8 points]accumulation.

10. Prove that every uncountable set of real numbers (bounded or not) must [5 points]contain a point that is a point of accumulation.


127

MATHEMATICS 242


March 10, 2000

1. (a) Define what is meant by a Cauchy sequence.

(b) Let {xn} be a sequence of real numbers with the property that

|xn+1 − xn| ≤ 12n

for all n = 1, 2, 3, . . . . Show that {xn} converges.

(c) Show that, for the sequence in (c),

x1 − 1 ≤ limn→∞

xn ≤ x1 + 1.

2. (a) Define what it means for a set G to be open.

(b) Prove or disprove that if G1, G2, G3, . . . are open sets then⋂m

k=1 Gk is openfor all positive integers m.

(c) Prove or disprove that if G1, G2, G3, . . . are open sets then⋂∞

k=1 Gk is open.

(d) Suppose that G is open and xn is a sequence converging to a point in G.Prove that there must be an integer m so that xk ∈ G for all k ≥ m.

3. Let f be a function defined on an interval [a, b] with the property thatfor all x ∈ [a, b] there is an integer p so that f(t) < p for all t ∈ [a, b] withx− 1/p < t < x + 1/p. Show that there is an integer P so that f(y) ≤ P for ally ∈ [a, b].



sup A = max A.




128





2xn


129

MATHEMATICS 242Final Examination

December 12, 2000Time: Three hours

NAME:I.D.:

Answer each of the true false questions on the examination sheet itself. Donot show any justifications

PART A. Each of these questions is either true or false. Circle the correctanswer. If it is true this means that you should be able to prove it; if it isfalse then you should have a counterexample in mind. You get one point forindicating correctly and you lose one point for an incorrect indication. Pleaseleave it blank if you are merely guessing or you may well guess yourself to anegative grade on this exam. [40 points]

1. True or False? For any set E ⊂ R if E is not closed then R \ E is notopen.

2. True or False? For any open, bounded set E ⊂ R the set E is compact.

3. True or False? If E is closed then no point of E can be an interior point.

4. True or False? If E is open then every point of E is an accumulationpoint.

5. True or False? If E is not closed then some point of E must be an interiorpoint.

6. True or False? If E is not open then some point of E must not be a pointof accumulation.

7. True or False? For any set E ⊂ R the set E′ is closed.

8. True or False? For any compact set E ⊂ R the set E′ is compact.




12. True or False? If E 6= E′ then E cannot be closed.

13. True or False? If E ⊂ E′ then E must be closed.

130


15. True or False? If E1, E2, . . . , Ep is a finite sequence of compact sets thenthe set




17. True or False? If E1, E2, . . . , is a sequence of compact sets then the set⋃∞i=1 Ei is also compact.


19. True or False? If A and B are open subsets of R then the set R \ (A∪B)is closed.

20. True or False? If A and B are open subsets of R then the set R \ (A∩B)is closed.


22. True or False? If x0 is an isolated point of a set A and A ⊂ B then x0





25. True or False? If f : [a, b] → R is a continuous function then there mustbe a point c ∈ [a, b] with the value f(c) = (f(a) + f(b))/2.

26. True or False? If f : R→ R is a continuous function then there is a t > 0so that |f(x)− f(y)| < 1 whenever |x− y| < t, x, y ∈ R.

27. True or False? If f : [a, b] → R is a continuous function then there is at > 0 so that |f(x)− f(y)| < 1 whenever |x− y| < t, x, y ∈ [a, b].

28. True or False? If f : R→ R is continuous at a point x0 then for all a > 0there is a b > 0 so that |f(x)− f(x0)| < a if |x− x0| < b.

29. True or False? If f : R→ R is continuous at a point x0 then for all c > 0there is a d > 0 so that |f(x)− f(x0)| < d if |x− x0| < c.

30. True or False? If f : R→ R is continuous and {xn} is a Cauchy sequencethen the sequence {f(xn)} must be convergent.

31. True or False? If f : R→ R is continuous and {xn} is a divergent sequencethen the sequence {f(xn)} must be also be divergent.

131

32. True or False? If limk→∞ xk = δ then for all a > 0 there is an integer pso that |xi − δ| < a whenever i > p.

33. True or False? If limk→∞ xk = δ then for all a > 0 there is an integer pso that |xi − δ| ≤ a whenever i ≥ p.

34. True or False? If limk→∞ xk = δ then for all a > 0 there is an integer Nso that |xk − a| < δ whenever k ≥ N .

35. True or False? If the sequence {an} is convergent then so too must be thesequence

bn =1

1 + an.

36. True or False? If the sequence {an} is divergent then so too must be thesequence

bn =1

1 + an.


bn =1

1 + an + (an)2.

38. True or False? Every sequence has a Cauchy subsequence.

39. True or False? Every bounded sequence has divergent subsequence.


132

Use separate EXAM BOOKLETS for parts B and C.PART B. Give complete proofs for each of the following statements if it is trueor give a counterexample if you believe it is false.

1. A sequence of real numbers is convergent if and only if it is a Cauchysequence. [10 points]

2. A function f : [a, b] → R that is continuous must be uniformly continuouson [a, b]. [10 points]

3. A function f : R→ R that is continuous must be uniformly continuous oneach interval (a, b) for a, b ∈ R, a < b. [3 points]

4. A function f : R→ R that is bounded and continuous must be uniformlycontinuous on R. [3 points]

PART C. Solve the following problems.

5. A sequence is defined recursively by a1 = 1 and an+1 = (an+1)/3. Determine [10 points]whether this sequence is bounded, monotonic, convergent and, if it is convergent,find its limit.

6. Let fn : [0, 1] → R be a sequence of functions such that each fn is monotonic [5 points]nondecreasing on [0, 1] and such that the limit

f(x) = limn→∞

fn(x)

exists for every x ∈ [0, 1]. Show that f is also monotonic nondecreasing on [0, 1].Would this assertion remain true if “monotonic nondecreasing” were changedin both cases to “monotonic increasing?”

7. Let f : [0, 1] → R be a continuous function. Show that there must be a [10 points]number ξ ∈ [0, 1] for which

∫ ξ

0

f(t) dt =∫ 1

ξ

f(t) dt.

133

MATHEMATICS 242

First Assignment

Due Friday, September 15Hand in to instructor mailbox or in class





2 is irrational.

4. (Counterexample) Prove or disprove: For any natural number n the equation4x2 + x− n = 0 has no rational root.


dn


134

MATHEMATICS 242

Final Assignment

Due December 4, 2000

1. If f is continuous and m ≤ f(x) ≤ M for all x in [a, b] show that

m(b− a) ≤∫ b

a

f(x) dx ≤ M(b− a).

2. Calculate∫ 1

0xp dx (for whatever values of p you can manage) by partitioning

[0, 1] into subintervals of equal length.

3. Calculate∫ b

axp dx (for whatever values of p you can manage) by partitioning

[a, b] into subintervals [a, aq], [aq, aq2], . . . [aqn−1, b] where aqn = b.

4. (Mean Value Theorem for Integrals) If f is continuous on [a, b] showthat there is a point ξ in (a, b) so that

∫ b

a

f(x) dx = f(ξ)(b− a).


∫ 1

0

f(x) dx =∫ 1

0

(f(x))2 dx.


2.

3. (a) (b)

(d)

(d)

135

MATHEMATICS 242

Second Assignment

Due September 15



2. Using the completeness axiom, show that every nonempty set E of realnumbers that is bounded below has a greatest lower bound (i.e., inf E existsand is a real number).




4. If a set E is dense, what can you conclude about a set A ⊃ E?

5. If a set E is dense, what can you conclude about the set R \ E?

6. If two sets E1 and E2 are dense, what can you conclude about the set E1∩E2?

136

MATHEMATICS 242

Third Assignment

Due October 6

1. Define

e = limn→∞

(1 +

1n

)n

assuming that this sequence does converge. Show that 2 < e < 3. Show that

limn→∞

(1 +

12n

)n

=√

e.


1 +2n

)=

(1 +

1n + 1

)(1 +

1n

)


limn→∞

(1 +

2n

)n

= e2.

2. A sequence {xn} is defined by starting at some value x1 = β and continuinginductively

xn+1 =x2

n − 2xn + 32

.

Discuss the convergence behaviour. (Hint: the cases β < 1, 1 < β < 3, etc.should be considered separately.)


limn→∞

an = limn→∞

sn.


imal element.


sup A = max A.


137







2xn


138

MATHEMATICS 242

Fourth Assignment

Due October 13 2000

1. Show that every bounded monotonic sequence is Cauchy without using themonotone convergence theorem.

2. Show that every Cauchy sequence is bounded without using the fact thatconvergent sequences must be bounded.





sup A = max A.








2xn


139

MATHEMATICS 242

Fifth Assignment

Due October 20 2000

1. Suppose that G is open and xn is a sequence converging to a point in G.Prove that there must be an integer m so that xk ∈ G for all k ≥ m. Show thatthis is not necessarily the case if G is closed but not open.













13. E′ is countably infinite.

14. E′ is uncountable.

15. E′ is nonempty but (E′)′= Ø.

140

MATHEMATICS 242

Sixth Assignment

October 20 2000


f(x) = inf{|x− e| : e ∈ E}.


{x ∈ IR : f(x) > 0} = (IR \ E).


f(x) > ε if x ∈ E ∩ (e− ε, e + ε).


f(e) > c



δ(A,B) = inf{|a− b| : a ∈ A, b ∈ B}.




4. Give an example of an open covering of the set CQ of rational numbers thatdoes not reduce to a finite subcover.

141

MATHEMATICS 242

Seventh Assignment


1. Prove the intermediate value theorem for continuous functions by using thefollowing “last point” argument: suppose that f(a) < 0 < f(b) and let z be thelast point in [a, b] where f(z) ≤ 0, i.e., let

z = sup{x ∈ [a, b] : f(x) ≤ 0}.

Show that f(z) = 0.

2. Show that the function f(x) = cos x defined for all x ∈ IR has a fixed pointα by applying the intermediate value property. Is α unique?

3. Let {xn} be defined by setting x1 = 1 and xn+1 = cos xn for all n.

(a) Show that if this is a convergent sequence then xn → α [same α as in #2].

(b) Show that there is a number 0 < β < 1 so that |xn−α| < βn. [Hint: use themean value theorem applied to the function f(x) = cos x on the interval [0, 1]and don’t forget that f(α) = α.]

(c) Now prove “again” that xn → α.

(d) For what value of N can you be sure that |xN − α| < 10−10?

(d) What is the best estimate for α that your calculator can give? How manytimes did you press the “cos” key?

142

MATHEMATICS 242

Ninth Assignment

November 24 2000

1. Let fn be a sequence of functions on an interval [a, b] and suppose that

f(x) = limn→∞

fn(x)

for each x ∈ [a, b].(a) If each fn is monotonic increasing does it follow that f is also monotonic

increasing?(b) If each fn is monotonic nondecreasing does it follow that f is monotonic

nondecreasing?(c) If each fn is bounded does it follow that f is bounded.(d) If the fn are uniformly bounded (meaning that there is a single number

M so that |fn(x)| ≤ M for all n and all x ∈ [a, b]) then does it follow that f isbounded?

(e) If each fn is continuous does it mean that f is continuous.(f) If the fn are uniformly equicontinuous (meaning that for every ε > 0 there

is a δ > 0 so that |fn(x) − fn(y)| < ε for all x, y ∈ [a, b] for which |x − y| < δand all n) then must f be uniformly continuous on [a, b]?

(g) If each fn is affine does it mean than f is affine?(h) If each fn is constant does it mean that f is constant?(i) If each fn is discontinuous somewhere does it mean that f must be

discontinuous somewhere?(j) If each fn is differentiable does it mean that f must be differentiable?(k) Is it necessarily true that

limx→a+

limn→∞

fn(x) = limn→∞

limx→a+

fn(x)?

(l) If each fn is nonnegative does it mean that f is nonnegative?(m) If each fn is unbounded does it mean that f is unbounded?

143

MATHEMATICS 831

Final Examination

December 1—12 2000

Time: Due on December 12, 2000.

Give complete solutions to ten of the following questions. You may use anytextbook or notes that you wish but you are not to consult with any individualsin person or electronically for assistance.

1. Discuss the validity of the following statements in which λ is Lebesgue outermeasure on the real line and all sets are sets of real numbers:

1. For any denumerable set E, λ(E) = 0.

2. For any open set G, λ(G) = λ(G).

3. For any closed set E with interior E0, λ(E) = λ(E0).

4. For any first category set E, λ(E) = 0.

5. For any dense set E and any interval (a, b),

λ(E ∩ (a, b)) = b− a.

6. For any set E,λ(E ∩ (0, 1)) = 1− λ((0, 1) \ E).

2. Let E1, E2, . . . , En be Lebesgue measurable subsets of [0, 1] such that eachpoint of [0, 1] belongs to at least q of these sets. Show that at least one of thesesets has measure greater than q/n.

3. Determine the Borel classification and also the Lebesgue measure of the set ofreal numbers in the interval [0, 1] that permit a decimal expansion that containsthe number 3.

4. If f is a continous real function then the set of points Ly = {x : f(x) = y} iscalled the level set for f at the level y. What is the largest cardinality possiblefor the set

{y : Ly has positive measure }?

5. Let E be a Lebesgue measurable set of positive measure and let xn be somesequence of points from the interval [0, 1]. Show that there must exist a pointy and a subsequence xnk

so that

y + xnk∈ E

for all k.

144

[Hint: Consider the functions fn(t) = χE(t − xn). Show that fn cannot beconverging almost everywhere to zero by considering the integrals of the fn.]

6. Let {fk} be a sequence of measurable functions defined on the real line andE a measurable set with finite measure. Suppose that

supk|fk(x)| < +∞

for every x ∈ E. Let ε > 0. Show that there is a closed set F ⊂ E and a finitenumber M so that |E \ F | < ε and |fk(x)| ≤ M for all k and all x ∈ F .

7. Let {ak} be a sequence of real numbers with∑∞

k=1 |ak| < +∞ and let {rk}be an enumeration of the rationals in [0, 1]. Show that

∞∑

k=1

ak√|x− rk|

converges absolutely almost everywhere in [0, 1].[Hint: What can you say about

∑∞k=1 |fk(x)| if

∑∞k=1

∫E|fk(x)| dx < +∞?]

8. Let f be a strictly positive Lebesgue integrable function on [a, b] and let0 < q ≤ b− a. Show that

infE∈S

∫

E

f(x) dx > 0

where S is the family of all Lebesgue measurable subsets E of [a, b] with m(E) ≥q.[Hint: If

∫ b

aχEn(t)f(t) dt tends to zero then some subsequence of . . . ]

9. Let f be Lebesgue integrable on [0, 1] and suppose that 0 < c < 1. If∫E

f(t) dt = 0 for every measurable set E ⊂ [0, 1] with m(E) = c then f mustvanish almost everywhere.

10. Characterize those functions f that are nonnegative and Lebesgue integrablein the interval [0, 1] and such that for every integer n = 1, 2, 3, 4, . . .

∫ 1

0

[f(x)]n dx =∫ 1

0

f(x) dx

11. Let f be Lebesgue integrable on the interval (−∞,∞) and let ε > 0.(a) Show that there is a continuous function g of compact support such

that ∫ ∞

−∞|f(x)− g(x)| dx < ε.

(b) Show that

limh→0

∫ ∞

−∞|f(x + h)− f(x)| dx = 0.

145

[Hint: For (a) apply Lusin’s theorem and for (b) use the approximation resultin (a).]

12. Prove that an arbitrary (uncountable) union of closed, nondegenerate in-tervals in one-dimension is a Borel set. Prove that an arbitrary union of closed,nondegenerate squares in two dimensions may not be a Borel set but is Lebesguemeasurable.[You can assume the Vitali theorem for squares and two dimensional Lebesguemeasure.]

13. Let X be a locally compact metric space, let B(X) be the class of Borel setsin X and let µ be a measure on B(X) such that µ(K) < ∞ for every compactsubset K of X. A system E of subsets of X is said to be upward directed iffor any two sets Eα, Eβ ∈ E there is a Eγ ∈ E with Eα ⊂ Eγ and Eβ ⊂ Eγ .Similarly a system E of subsets of X is said to be downward directed if for anytwo sets Eα, Eβ ∈ E there is a Eγ ∈ E with Eα ⊃ Eγ and Eβ ⊃ Eγ .

Prove that ifµ(

⋃α

Gα) = supα

µ(Gα)

for every upward directed system {Gα} of open sets then

µ(⋂α

Kα) = infα

µ(Kα)

for every downward directed system of compact sets.(Hint: Show that every compact subset of X is contained in an open set

with compact closure.)

14. Let f be a measurable real function, A a measurable set, let λ denoteLebesgue measure on the line and write

w(y) = λ({x ∈ A : |f(x)| > y}).

Show that w is continuous if and only if λ(f−1(y)) = 0 for every real number y,that

w(y) ≤ 1y

∫

A

|f(x)| dx,

for every real number y, and that∫

A

|f(x)| dx =∫ ∞

0

w(y)dy.

146

MATHEMATICS 242


October 4 2000

There are four questions. Answer all four.

1. Define what the statement

limk→∞

xk = z

means for a sequence of real numbers x1, x2, x3, . . . and a real number z.

2. Using your definition in #1 and no other theorems of the course, prove thatif limk→∞ xk = z then

limk→∞

f (xk) = f(z)

for any affine function5 f : R→ R.

3. Let S ⊂ R, let f : R→ R be an affine function and define the image set

f(S) = {f(s) : s ∈ S}.

(a) Prove or disprove: If S is countable then f(S) is also countable.

(b) Prove or disprove: If S is uncountable then f(S) is also uncountable.

(c) Prove or disprove: If S is dense then f(S) is also dense.

(d) Prove or disprove: If S is bounded then f(S) is also bounded.

4. Let β ∈ R and define a sequence recursively by writing

x1 = β , xn+1 =x2 + x + 1

5for n = 1, 2, 3, . . .

(i) Let β = 0. Prove inductively that the sequence {xn} is increasing. Does itconverge and if so to what does it converge?

(ii) Let β = 4. Prove inductively that the sequence {xn} is increasing. Does itconverge and if so to what does it converge?

5A function f : R → R is said to be affine if it is of the form f(x) = αx + β for α, β ∈ Rand α 6= 0. Sometimes such functions are (incorrectly) called “linear.”

147

MATHEMATICS 242


October 4 2000


1. Define what the statement

limk→∞

xk = z

means for a sequence of real numbers x1, x2, x3, . . . and a real number z.

Just use textbook definition.

2. Using your definition in #1 and no other theorems of the course, prove thatif limk→∞ xk = z then

limk→∞

f (xk) = f(z)

for any affine function6 f : R→ R.

Suppose that f(x) = αx + β where α, β ∈ R and α 6= 0. Let ε > 0and choose N so that |xn − z| < ε/|α| whenever n ≥ N .

Then

|f(xn)− f(z)| = |(αxn + β)− (αz + β)| = |α| |xn − z| < ε

whenever n ≥ N . QED

3. Let S ⊂ R, let f : R→ R be an affine function and define the image set

f(S) = {f(s) : s ∈ S}.

(a) Prove or disprove: If S is countable then f(S) is also countable.

To begin with, for this and the other three problems let f(x) = αx+βand let g be the inverse function (you can compute it explicitly ifyou wish g(x) = (x− β)/α but it is not needed really).

If S is countable then it is the range of some sequence s1, s2, s3, . . .and then it is clear that f(S) is the range of the sequence

f(s1), f(s2), f(s3), . . .

and so also countable.

[Note that this works for any function f , not just for affine functionsor one-one functions.]

6A function f : R → R is said to be affine if it is of the form f(x) = αx + β for α, β ∈ Rand α 6= 0. Sometimes such functions are (incorrectly) called “linear.”

148

(b) Prove or disprove: If S is uncountable then f(S) is also uncountable.

If not then f(S) is countable and it follows from part (a) that g(f(S))must be countable too. But g(f(S)) = S and so this is a contradic-tion.

(c) Prove or disprove: If S is dense then f(S) is also dense.

WLOG we can assume that α > 0 since the case α < 0 can besimilarly handled. Note that this means f and g are both increasing.

Let (a, b) be any interval. The interval (g(a), g(b)) must contain apoint of S since S is dense. Thus there is a point s1 ∈ S with g(a) <s1 < g(b). Since f is increasing we have f(g(a)) < f(s1) < f(g(b))or, more simply,

a < f(s1) < b

and this exhibits a point in f(S) inside the interval (a, b). Thus f(S)is dense as required.

(d) Prove or disprove: If S is bounded then f(S) is also bounded.

If S is bounded then there is a real number M with |s| ≤ M for alls ∈ S. This means

|f(s)| = |αs + β| ≤ |αM |+ |β|

for all s ∈ S, i.e., that |αM |+ |β| is an upper bound for f(S). QED

4. Let β ∈ R and define a sequence recursively by writing

x1 = β , xn+1 =x2 + x + 1

5for n = 1, 2, 3, . . .

(i) Let β = 0. Prove inductively that the sequence {xn} is increasing. Does itconverge and if so to what does it converge?

First let’s find the fixed points of the function F (x) = x2+x+15 . Solv-

ingx2 + x + 1

5= x

gives two roots c = 2−√3 and d = 2+√

3. Remember that F (c) = cand F (d) = d.

To show that the sequence increases check first that x1 < x2 (i.e.,that 0 < 1/5).

149

Then assume that xn < xn+1 for some integer n. It is easy to checkthat

xn+1 = F (xn) < F (xn+1) = xn+2

now follows (either note that F is increasing or just do the inequal-ities). It follows by induction that the sequence is increasing.

Now check an upper bound. Claim xn < c for all n. Certainlyx1 < c. Assume that xn < c for some value of n. Then xn+1 =F (xn) < F (c) = c follows (again because F is increasing or elsejust check the inequality directly). It follows by induction that thesequence is bounded above by c.

Consequently the sequence converges. To what? To either c or dbut evidently only the former is possible. QED.

(ii) Let β = 4. Prove inductively that the sequence {xn} is increasing. Does itconverge and if so to what does it converge?

To check increasing note that x1 = 4 and x2 > 4. The inductionstep is exactly the same as for part (i) and need not be repeated. Itfollows by induction that the sequence is increasing.

If it converges then it converges to c or d neither of which is possiblesince c < d < 4 < xn for all n. Thus the sequence diverges.

150

MATHEMATICS 242


November 8 2000


1. Define what the following terms mean: interior point, accumulation point,[1 pt]isolated point, open set, and closed set.

2. Using your definitions in #1 and no other theorems of the course, prove that[3 pt]if G is an open set then the complementary set F = R \G must be closed.

3. The following statement is a true theorem: If E ⊂ R is a compact set suchthat each of its points is isolated in E then E is finite.

(a) Prove this theorem using the fact that every compact set has the Bolzano-[3 pt]Weierstrass property.

(b) Prove this theorem using the fact that every compact set has the Heine-[3 pt]Borel property.

(c) Prove this theorem using a “last point” argument: Define[3 pt]

S = {s : (−∞, s) ∩ E is finite}.

Show that sup S cannot be −∞. Show that sup S cannot be finite. Concludethe theorem.

(d) Prove or disprove: If E ⊂ R is a closed set such that each of its points is[1 pt]isolated in E then E is finite.

(e) Prove or disprove: If E ⊂ R is a bounded set such that each of its points is[1 pt]isolated in E then E is finite.

4. Let f : [a, b] → [a, b] be continuous (i.e., f is a continuous function on [a, b]with range contained also in [a, b]).

(a) Show that f must have a fixed point (i.e., that there is at least one point ξ[2 pt]in [a, b] for which f(ξ) = ξ).

(b) Give an example to illustrate that the fixed point need not be unique.[1 pt]

(c) Show that the set of all fixed points of f is necessarily closed.[2 pt]

151

MATHEMATICS 242


November 8 2000


1. Define what the following terms mean: interior point, accumulation point,[1 pt]isolated point, open set, and closed set.

Just use textbook definitions.

2. Using your definitions in #1 and no other theorems of the course, prove that[3 pt]if G is an open set then the complementary set F = R \G must be closed.

Again textbook material.

3. The following statement is a true theorem: If E ⊂ R is a compact set suchthat each of its points is isolated in E then E is finite.

(a) Prove this theorem using the fact that every compact set has the Bolzano-[3 pt]Weierstrass property.

If E is not finite then there is a sequence {xk} of distinct elementsof E. By the BW property there is a convergent subsequence {xkp}converging to a point z ∈ E. Evidently z is an accumulation pointof E and not isolated, in contradiction to the hypothesis.

(b) Prove this theorem using the fact that every compact set has the Heine-[3 pt]Borel property.

For each x ∈ E there is an open interval Ix so that Ix contains nopoint of E other than x (this is because each point of E is isolated).By the HB property there is a finite subcover Ix1 , Ix2 , . . . Ixm of E.This means E contains at most m elements.

(c) Prove this theorem using a “last point” argument: Define[3 pt]

S = {s : (−∞, s) ∩ E is finite}.

Show that sup S cannot be −∞. Show that sup S cannot be finite. Concludethe theorem.

152

Since E is bounded, let a = inf E and note that a ∈ S. Thussup S 6= −∞.

If z = sup S is finite then for any ε > 0 the point z−ε is not an upperbound of S while z + ε cannot belong to S. Thus S ∩ (−∞, z − ε) isfinite while S ∩ (−∞, z + ε) is infinite. In particular S ∩ (z− ε, z + ε)is infinite and so z is a point of accumulation of E. But E is closedand z ∈ E and z is not an isolated point. Since this is impossible,sup s = ∞.

Take any b > sup E and note that b ∈ S so that E ⊂ E ∩ (−∞, b) isfinite, as required.

(d) Prove or disprove: If E ⊂ R is a closed set such that each of its points is[1 pt]isolated in E then E is finite.

For a counterexample, let E be the (infinite and closed ) set of allintegers, all of whose points are isolated.

(e) Prove or disprove: If E ⊂ R is a bounded set such that each of its points is[1 pt]isolated in E then E is finite.

For a counterexample, let E be the (infinite and bounded) set of allfractions of the form 1/n for n a positive integer, all of whose pointsare isolated.

4. Let f : [a, b] → [a, b] be continuous (i.e., f is a continuous function on [a, b]with range contained also in [a, b]).

(a) Show that f must have a fixed point (i.e., that there is at least one point ξ[2 pt]in [a, b] for which f(ξ) = ξ).

Let g(x) = f(x) − x. Certainly g is continuous, since f is, andg(a) = f(a)− a ≥ 0 while g(b) = f(b)− b ≤ 0. By the IVP propertyof continuous functions there is a point ξ ∈ [a, b] for which g(ξ) = 0.

(b) Give an example to illustrate that the fixed point need not be unique.[1 pt]

Take f(x) = x for example.

(c) Show that the set of all fixed points of f is necessarily closed.[2 pt]

Let E = {x ∈ [a, b] : f(x) = x}. If z is a point of accumulation ofE then some sequence xn → z for xn ∈ E. But by the continuityof f , f(xn) → f(z) so f(xn) = xn implies f(z) = z. Thus z ∈ E.Consequently E is closed (it contains all points of accumulation).

153

MATHEMATICS 831

Final Examination

December 1—12 2000

Time: Due on December 12, 2000.

Give complete solutions to ten of the following questions. You may use anytextbook or notes that you wish but you are not to consult with any individualsin person or electronically for assistance.

1. Discuss the validity of the following statements in which λ is Lebesgue outermeasure on the real line and all sets are sets of real numbers:

1. For any denumerable set E, λ(E) = 0.

2. For any open set G, λ(G) = λ(G).

3. For any closed set E with interior E0, λ(E) = λ(E0).

4. For any first category set E, λ(E) = 0.

5. For any dense set E and any interval (a, b),

λ(E ∩ (a, b)) = b− a.

6. For any set E,λ(E ∩ (0, 1)) = 1− λ((0, 1) \ E).

2. Let E1, E2, . . . , En be Lebesgue measurable subsets of [0, 1] such that eachpoint of [0, 1] belongs to at least q of these sets. Show that at least one of thesesets has measure greater than q/n.

3. Determine the Borel classification and also the Lebesgue measure of the set ofreal numbers in the interval [0, 1] that permit a decimal expansion that containsthe number 3.

4. If f is a continous real function then the set of points Ly = {x : f(x) = y} iscalled the level set for f at the level y. What is the largest cardinality possiblefor the set

{y : Ly has positive measure }?

5. Let µ∗ be an outer measure on a space X. Show that a function f : X → IRis measurable if and only if for every T ⊂ X and all −∞ < a < b < ∞

µ∗(T ) ≥ µ∗({x ∈ T : f(x) ≤ a}) + µ∗({x ∈ T : f(x) ≥ b}).

6. Let E be a Lebesgue measurable set of positive measure and let xn be some

154

sequence of points from the interval [0, 1]. Show that there must exist a pointy and a subsequence xnk

so that

y + xnk∈ E

for all k.

7. Let {fk} be a sequence of Lebesgue measurable functions defined on the realline and E a Lebesgue measurable set with finite measure. Suppose that

supk|fk(x)| < +∞

for every x ∈ E. Let ε > 0. Show that there is a closed set F ⊂ E and a finitenumber M so that λ(E \ F ) < ε and |fk(x)| ≤ M for all k and all x ∈ F .

8. Let {ak} be a sequence of real numbers with∑∞

k=1 |ak| < +∞ and let {rk}be an enumeration of the rationals in [0, 1]. Show that

∞∑

k=1

ak√|x− rk|

converges absolutely almost everywhere in [0, 1].

9. Let f be Lebesgue integrable on [0, 1] and suppose that 0 < c < 1. If∫E

f(t) dt = 0 for every measurable set E ⊂ [0, 1] with λ(E) = c then f mustvanish almost everywhere.

10. Characterize those functions f that are nonnegative and Lebesgue integrablein the interval [0, 1] and such that for every integer n = 1, 2, 3, 4, . . .

∫ 1

0

[f(x)]n dx =∫ 1

0

f(x) dx.

11. Let f be Lebesgue integrable on the interval (−∞,∞) and let ε > 0.(a) Show that there is a continuous function g of compact support such

that ∫ ∞

−∞|f(x)− g(x)| dx < ε.

(b) Show that

limh→0

∫ ∞

−∞|f(x + h)− f(x)| dx = 0.

12. Let X be a locally compact metric space, let B(X) be the class of Borel setsin X and let µ be a measure on B(X) such that µ(K) < ∞ for every compactsubset K of X. A system E of subsets of X is said to be upward directed iffor any two sets Eα, Eβ ∈ E there is a Eγ ∈ E with Eα ⊂ Eγ and Eβ ⊂ Eγ .

155

Similarly a system E of subsets of X is said to be downward directed if for anytwo sets Eα, Eβ ∈ E there is a Eγ ∈ E with Eα ⊃ Eγ and Eβ ⊃ Eγ .

Prove that ifµ(

⋃α

Gα) = supα

µ(Gα)

for every upward directed system {Gα} of open sets then

µ(⋂α

Kα) = infα

µ(Kα)

for every downward directed system of compact sets.(Hint: Show that every compact subset of X is contained in an open set

with compact closure.)

156


December 12, 2000Time: Three hours

NAME:I.D.:

Answer each of the true/false questions on the examination sheet itself. Donot show any justifications

PART A. Each of these questions is either true or false. CIRCLE the correctanswer. If it is true this means that you should be able to prove it; if it isfalse then you should have a counterexample in mind. You get one point forindicating correctly and you lose one point for an incorrect indication. Pleaseleave it blank if you are merely guessing or you may well guess yourself to anegative grade on this exam.[35 points]

1. True or False? For any set E ⊂ R if E is not closed then R \ E is notopen.

2. True or False? For any open, bounded set E ⊂ R the set E is compact.

3. True or False? If E is closed then no point of E can be an interior point.

4. True or False? If E is open then every point of E is an accumulationpoint.

5. True or False? If E is not closed then some point of E must be an interiorpoint.

6. True or False? If E is not open then some point of E must not be a pointof accumulation.

7. True or False? For any set E ⊂ R the set E′ is closed.

8. True or False? For any compact set E ⊂ R the set E′ is compact.




12. True or False? If E 6= E′ then E cannot be closed.

13. True or False? If E ⊂ E′ then E must be closed.


157





17. True or False? If E1, E2, . . . , is a sequence of compact sets then the set⋃∞i=1 Ei is also compact.


19. True or False? If A and B are open subsets of R then the set R \ (A∪B)is closed.

20. True or False? If A and B are open subsets of R then the set R \ (A∩B)is closed.


22. True or False? If x0 is an isolated point of a set A and A ⊂ B then x0





25. True or False? If f : [a, b] → R is a continuous function then there mustbe a point c ∈ [a, b] with the value f(c) = (f(a) + f(b))/2.

26. True or False? If f : R→ R is a continuous function then there is a t > 0so that |f(x)− f(y)| < 1 whenever |x− y| < t, x, y ∈ R.

27. True or False? If f : [a, b] → R is a continuous function then there is at > 0 so that |f(x)− f(y)| < 1 whenever |x− y| < t, x, y ∈ [a, b].

28. True or False? If f : R→ R is continuous and {xn} is a Cauchy sequencethen the sequence {f(xn)} must be convergent.

29. True or False? If f : R→ R is continuous and {xn} is a divergent sequencethen the sequence {f(xn)} must be also be divergent.


bn =1

1 + an.

158

31. True or False? If the sequence {an} is divergent then so too must be thesequence

bn =1

1 + an.


bn =1

1 + an + (an)2.

33. True or False? Every sequence has a Cauchy subsequence.

34. True or False? Every bounded sequence has divergent subsequence.


159

Use separate EXAM BOOKLETS for parts B and C.

PART B. Give complete proofs for each of the following statements if it is trueor give a counterexample if you believe it is false.

1. A sequence of real numbers is convergent if and only if it is a Cauchysequence.[15 points]

2. A function f : [a, b] → R that is continuous must be uniformly continuouson [a, b].[15 points]

PART C. Solve the following problems.

3. A sequence is defined recursively by a1 = 1 and an+1 = (an+1)/3. Determine[10 points]whether this sequence is bounded, monotonic, convergent and, if it is convergent,find its limit.

4. Let fn : [0, 1] → R be a sequence of functions such that each fn is monotonic[10 points]nondecreasing on [0, 1] and such that the limit

f(x) = limn→∞

fn(x)

exists for every x ∈ [0, 1]. Show that f is also monotonic nondecreasing on [0, 1].Would this assertion remain true if “monotonic nondecreasing” were changedin both cases to “monotonic increasing?”

5. Let f : [0, 1] → R be a nonnegative, continuous function. Show that there[10 points]must be a number ξ ∈ (0, 1) for which

∫ ξ

0

f(t) dt =∫ 1

ξ

f(t) dt.

160

MATHEMATICS 320

Final Assignment

Due November 29 1993

1. Establish the inequality∣∣∣∣∣∫ b

a

f(x)g(x) dx

∣∣∣∣∣ ≤(∫ b

a

f2(x)) dx

) 12

(∫ b

a

g2(x) dx

) 12

for functions f and g integrable on [a, b].

2. A function is Lipschitz of order α on an interval [a, b] provided

|f(x)− f(y)| ≤ M |x− y|α

for some constant M and all x, y∈ [a, b]. Show that a continuously differentiablefunction is Lipschitz of order 1. Give an example to show that the converse isnot true.

Explain why Lipschitz functions of order α > 1 are of little interest.

3. Let f be 2π–periodic and k times continuously differentiable. Show that theFourier coefficients of f satisfy

|an| ≤ C

nkand |bn| ≤ C

nk

for some constant C.

4. Let f be an integrable function on [−π, π], let sn(x) denote the partial sumsof its Fourier series and and let σn(x) denote the Cesaro averages of these. Ifm ≤ f(x) ≤ M for all x then show that

m ≤ σn(x) ≤ M

for all x and n but thatm ≤ sn(x) ≤ M

need not hold.

5. Show thatπ2

8= 1 +

132

+152

+172

+192

. . .

andπ4

96= 1 +

134

+154

+174

+194

. . . .

[Hint: consider the Fourier series for |x| on [−π, π].]

161

MATHEMATICS 320

First Assignment

Dated September 15 1993

Prove the following assertions about the upper and lower limits of a sequenceof real numbers.

1. If a sequence {an} converges to a real number L then

lim supn→∞

an = lim infn→∞

an = L.

2. If a sequence {an} diverges to +∞ then

lim supn→∞

an = lim infn→∞

an = +∞.

3. If lim supn→∞ an = L for a finite real number L and ε > 0 then

an > L + ε

for only finitely many n andan > L− ε

for infinitely many n.

4. Give examples of sequences of rational numbers {an} with (i) upper limit√2 and lower limit −√2, (ii) upper limit +∞ and lower limit

√2, (iii) upper

limit π and lower limit e.

5. Show that for any sequences {an} and {bn}

lim supn→∞

(an + bn) ≤ lim supn→∞

an + lim supn→∞

bn.

Give an example to show that the inequality

lim supn→∞

(an + bn) < lim supn→∞

an + lim supn→∞

bn.

may occur.

162

MATHEMATICS 320

Second [graded] Assignment

Due October 1, 1993

1. Prove the following assertion about the upper and lower limits for anysequence {an} of positive real numbers:

lim infn→∞

an+1

an≤ lim inf

n→∞n√

an ≤ lim supn→∞

n√


an+1

an.

Give an example to show that each of these inequalities may be strict.Explain why these inequalities show that the root test is more powerful

(formally) than the ratio test.

2. Show thatπ

4≤

∞∑

k=1

1k2 + 1

≤ π

4+ 1.

163

MATHEMATICS 320

Third Assignment

Due October 8, 1993


(1

1− x

)2

=∞∑

n=0

(n + 1)xn

for appropriate values of x.

2. Show that the formal product of the series( ∞∑

n=0

xn

√n + 1

) ( ∞∑n=0

xn

√n + 1

)

converges absolutely for |x| < 1 and diverges if x = −1.

3. Show that for any finite sequences {a1, a2, . . . , am} and {b1, b2, . . . , bm} theinequality ∣∣∣∣∣

n∑

k=1

akbk

∣∣∣∣∣ ≤(

n∑

k=1

(ak)2) 1

2(

n∑

k=1

(bk)2) 1

2

must hold.

4. If {an} is a sequence of nonnegative numbers for which∑∞

n=1 an convergesthen the series ∞∑

n=0

√an

n

also converges.

164

MATHEMATICS 320

Fourth Assignment

Due October 22 1993

1. If {fn} and {gn} converge uniformly on a set E then so too does the sequence{fn + gn}.

2. If {fn} converges uniformly on a set E to a function f and each fn is boundedon E then so too is f .

3. If {fn} and {gn} are sequences of bounded functions that converge uniformlyon a set E then so too does the sequence {fngn}.

4. Prove that this statement need not be true: if {fn} and {gn} convergeuniformly on a set E then so too does the sequence {fngn}.

5. If {fn} is a sequence of continuous functions converging uniformly on aninterval (a, b) and {xn} is a sequence of numbers converging to a point x ∈ (a, b)then

limn→∞

fn(xn) = f(x).


(−1)n x2 + n

n2

converges uniformly on every bounded interval but does not converge absolutelyat any value of x.

165

MATHEMATICS 320

Fifth Assignment

Due Wednesday, Nov 3, 1993

1. Let the series∑∞

n=1 an converge and suppose that bn is a decreasing sequenceof positive numbers. Show that the series

∑∞n=1 anbn also converges.

2. If∑∞

n=1 anxn converges uniformly on (−1, 1) then it converges uniformly on[−1, 1].

3. If∑∞

n=1 n|bn| < +∞ and

f(x) =∞∑

n=1

bn sin nx

then

f ′(x) =∞∑

n=1

nbn cosnx

for all x.

4. State conditions under which the identity

∞∑n=0

anxn

1− x=

∞∑n=0

(a0 + a1 + . . . an)xn

is valid.

5. Find a closed form expression for the function

∞∑n=0

(1 +

11!

+12!

+ . . .1n!

)xn.

166

MATHEMATICS 320


October 13, 1993


lim supn→∞


an + lim supn→∞

bn.


lim supn→∞


an + lim supn→∞

bn.

may occur.

2. Let {xn} be a convergent sequence. Determine the behaviour of the series

x0 + (x1 − x0) + (x2 − x2) + (x3 − x2) + . . . .

From this and the identity

1n(n + 1)

=1n− 1

n + 1

determine the sum of

11 · 2 +

12 · 3 +

13 · 4 +

14 · 5 + . . . .

3. Let an be a decreasing sequence of positive numbers. Is the following a validtest?

∑∞n=1 an is convergent if and only if limnan = 0.

(Is it a necessary condition? Is it a sufficient condition?)

167

MATHEMATICS 320


November 10 1993

1. If fn is a sequence of continuous functions on the interval [0, 1] and {fn}converges uniformly on (0, 1) then it must also converge uniformly on [0, 1].

2. If fn is a sequence of functions on the interval [0, 1] and {fn} convergesuniformly on (0, 1) then it need not converge uniformly on [0, 1].

3. If fn and gn are sequences of continuous functions on the interval [0, 1] and{fn} and {gn} converge uniformly on [0, 1] then {fngn} must also convergeuniformly on [0, 1].

4. If fn and gn are sequences of functions on the interval [0, 1] and {fn} and{gn} converge uniformly on [0, 1] then {fngn} need not converge uniformly on[0, 1].

168

MATHEMATICS 320

First Assignment

Due September 15

Some information about the course:You can reach the instructor at 291-4233 or [email protected] or during officehours in Room 10,519 on MWF 11:30–12:30. The grading for the course is basedon 10% homework, 20% for each of two midterm examinations and 50% for thefinal three hour examination.

Hand in answers to #4 and #5 only.



lim supn→∞

an = lim infn→∞

an = L.


lim supn→∞

an = lim infn→∞

an = +∞.


an > L + ε




√2, (iii) upper


5. Show that for any sequences {an} and {bn}lim sup

n→∞(an + bn) ≤ lim sup

n→∞an + lim sup

n→∞bn.


lim supn→∞


an + lim supn→∞

bn.

may occur. What are the corresponding assertions for the limit inferior?

169

MATHEMATICS 320

Second Assignment

Due September 22 1995


lim infn→∞

an+1

an≤ lim inf

n→∞n√


n√


an+1

an.



2. Show thatπ

4≤

∞∑

k=1

1k2 + 1

≤ π

4+ 1.

170

MATHEMATICS 320

Third Assignment


1. Let {an} be a decreasing sequence of positive numbers. Then∑

an convergesonly if lim nan = 0. Does the converse hold?

2. Let {an} be a sequence of positive numbers and write

Ln =log

(1

an

)

log n.

Show that if lim inf Ln > 1 then∑

an converges. Show that if Ln ≤ 1 for allsufficiently large n then

∑an diverges.

3. Apply the test in #2 to obtain convergence or divergence of the followingseries (x is positive):

∞∑n=2

xlog n∞∑

n=2

xlog log n∞∑

n=2

(log n)− log n

4. Let F be a continuous, differentiable, positive function on [1,∞) with apositive, decreasing derivative F ′. Show that

∑F ′(i) converges if and only if∑

F ′(i)/F (i) converges.Suppose that

∑F ′(i) diverges. Show that

∑F ′(i)/[F (i)]p converges if and

only if p > 1.

171

MATHEMATICS 320

Fourth Assignment

Due October 2 1995


x0 + (x1 − x0) + (x2 − x2) + (x3 − x2) + . . . .

From this and the identity 1n(n+1) = 1

n − 1n+1 determine the sum of

11 · 2 +

12 · 3 +

13 · 4 +

14 · 5 + . . . .


n=0

xn

√n + 1

) ( ∞∑n=0

xn

√n + 1

)

converges absolutely for |x| < 1 and diverges if x = −1.3. Show that for any finite sequences {a1, a2, . . . , am} and {b1, b2, . . . , bm} theinequality ∣∣∣∣∣

n∑

k=1

akbk

∣∣∣∣∣ ≤(

n∑

k=1

(ak)2) 1

2(

n∑

k=1

(bk)2) 1

2

must hold.4. If {an} is a sequence of nonnegative numbers for which

∑∞n=1 an converges

then the series ∞∑n=0

√an

n

also converges.

172

MATHEMATICS 320

Fifth Assignment

Due October 23, 1995






limn→∞

fn(xn) = f(x).


(−1)n x2 + n

n2


173

MATHEMATICS 320

Sixth Assignment

Due October 27, 1993

1. If∑∞


2. If∑∞

n=1 n|bn| < +∞ and

f(x) =∞∑

n=1

bn sin nx

then

f ′(x) =∞∑

n=1

nbn cosnx

for all x.


∞∑n=0

anxn

1− x=

∞∑n=0

(a0 + a1 + . . . an)xn

is valid.


∞∑n=0

(1 +

11!

+12!

+ . . .1n!

)xn.

174

MATHEMATICS 320

Seventh Assignment

Due November 4 1995

1. Let f(x) =∑∞

n=0 anxn with each an ≥ 0 and so that the series has radius ofconvergence 1. If limx→1− f(x) = L then

∑∞n=0 an = L.

2. Give an example of a series f(x) =∑∞

n=0 anxn with radius of convergence 1so that limx→1− f(x) = L exists and yet

∑∞n=0 an diverges.

3. Definef(x) = 1 + x + x2/2! + x3/3! + . . . xn/n! + . . . .

Obtain several interesting properties of this function just using power seriesmethods.

175

MATHEMATICS 320

Eighth Assignment

Due November 11 1995

1. Show that if the integral∫∞0

f(x) dx converges then so also does the series

∞∑n=1

∫ n

n−1

f(x) dx

but that the converse statement is not true.

2. Prove a version of the ratio test for infinite integrals: Suppose f is continuouson [1,∞) and

lim supx→∞

∣∣∣∣f(x + 1)

f(x)

∣∣∣∣ < 1

then the integral∫∞1

f(x) dx is absolutely convergent.

3. Prove a version of the root test for infinite integrals: Suppose f is continuouson [1,∞) and

lim supx→∞

∣∣∣f(x)1/x∣∣∣ < 1

then the integral∫∞1

f(x) dx is absolutely convergent.

4. Show that the integral ∫ ∞

0

tne−xt2 dt

is uniformly convergent on [a,∞) for each a > 0. Show that∫ ∞

0

t2ne−xt2 dt =12Γ(n + 1/2)x−n−1/2.

176

MATHEMATICS 320

Final Assignment

Due December 5 1995


m ≤ σn(x) ≤ M


need not hold.

2. Show thatπ2

8= 1 +

132

+152

+172

+192

. . .

andπ4

96= 1 +

134

+154

+174

+194

. . . .


3. Let f be a continuous function on the interval [0, 1] and suppose that forevery n = 0, 1, 2, 3, . . . ∫ 1

0

f(t)tn dt = 0.

What can you conclude?[Hint: Use the Weierstrass approximation theorem and argue using inte-

grals.]

4. Let f and g be piecewise continuous functions on [−π, π] with Fourier coef-ficients an, bn and αn, βn respectively. Show that

1π

∫ π

−π

f(t)g(t) dt = 2a0α0 +∞∑

k=1

anαn + bnβn.

5. Establish the inequality

(∫ b

a

(f(x) + g(x))2 dx

) 12

≤(∫ b

a

f2(x)) dx

) 12

+

(∫ b

a

g2(x) dx

) 12


177

MATHEMATICS 320


October 13 1995

1. Prove that the series ∞∑2

1n(lnn)p

converges for all p > 1 and diverges for all 0 < p ≤ 1 by using either the integraltest or the Cauchy condensation test.

2. Let an be a sequence of postive numbers and write

Mn =ln (1/nan)ln(lnn)

.

(i). Show that if lim infn→∞Mn > 1 then∑∞

k=1 ak converges.

(ii). Show that if Mn ≤ 1 for all sufficiently large n then∑∞

k=1 ak diverges.

3. Show that the series ∞∑

k=1

n[1+1/ln n]

diverges.

178

MATHEMATICS 320


November 17 1995

1. Define what is meant by a sequence of functions {fn} converging uniformlyto a function f on a set E.

Prove or disprove: if a sequence of functions {fn} converges uniformly to afunction f on a set E then {(fn)2} converges uniformly to the function f2 onE.

Prove or disprove: if a sequence of functions {fn} converges uniformly to afunction f on a set E then {|fn|} converges uniformly to the function |f | on E.

2. From the power series

(1 + x2)−1 =∞∑0

(−1)kx2k

give all the necessary justifications (convergence, uniform convergence etc.) toobtain by integration that

π/4 =∞∑0

(−1)k1/(2k + 1).

3. Let g be a continuous function on [0,∞) so that for some p ≤ 0

lim supx→∞

|xpg(x)| < ∞

and define the function

F (x) =∫ ∞

0

e−xtg(t) dt.

Show that F is infinitely differentiable.

179

MATHEMATICS 320

Final Examination

December 4 1997


Ln =log (1/an)

log n.

Show that iflim infn→∞

Ln > 1

then∑

an converges. Show that if Ln ≤ 1 for all sufficiently large n then∑

an

diverges.

2. What assertions about the series∑

an can be made if you are given that

lim supn→∞

log (1/an)log n

> 1?

3. Explain how the test in question #1 can be used to help determine the setof values of x for which the series

∞∑n=2

xlog n

converges. What is the exact set of convergence?

4. What is the best assertion you are able to make about the uniform conver-gence of

∞∑n=2

xlog n?

5. What is the best assertion you are able to make about the value

limx→α+

∞∑n=2

xlog n?

6. What is the best assertion you are able to make about the value of

∫ b

a

( ∞∑n=2

xlog n

)dx?

180


d

dx

( ∞∑n=2

xlog n

)?

8. A sequence of functions {fn} defined on an interval I is said to convergecontinuously to the function f if fn(xn) → f(x0) whenever {xn} is a sequenceof points in the interval I that converges to a point x0 in I. Prove the followingtheorem:

Let {fn} be a sequence of continuous functions on an interval [a, b].Then {fn} converges continuously on [a, b] if and only if {fn} con-verges to f uniformly on [a, b].

Does the theorem remain true if the interval [a, b] is replaced with (a, b) or[a,∞)?

You would be expected to complete at least five of these questions in the three hours of the

examination period. If you are unable to complete any of the questions in the time allowed write

up a solution and submit to me or my mailbox by Friday at 4:00pm.

181

MATHEMATICS 320


October 1 1997

1. Let σ : I → I one-one and onto. Show that if the infinite sum∑

i∈I

ai

converges then so too does the infinite sum∑

i∈I

aσ(i).

2. Let {xk} be a sequence of positive numbers. Consider the following condi-tions:(a) lim sup

k→∞

√kak > 0

(b) lim supk→∞

√kak < ∞

(c) lim infk→∞

√kak > 0

(d) lim infk→∞

√kak < ∞

Which condition(s) imply convergence or divergence? Supply proofs.Which conditions are inconclusive as to convergence or divergence? Supply

examples.

If you are unable to complete any of the questions in the time allowed writeup a solution and submit it in Friday’s class.

182

MATHEMATICS 320


November 5 1997

1. For what values of x do the series converge?(i)

1 + αx + βx2 + α2x3 + β2x4 + α3x5 + β3x6 + . . .

(ii)1 + αx + αβx2 + α2βx3 + α2β2x4 + α3β2x5 + α3β3x6 + . . .

Here α and β are positive constants.

2. For what values of x does the product

∞∏

k=1

(1 +

x2k

k2

)

converge?

3. Show that A ∪ B must be of measure zero if both A and B are sets of realnumbers of measure zero.

4. Directly from the definition of the integral prove that∫ 1

0

f(x) dx =∫ 0

−1

f(−x) dx

for any continuous function f on the interval [0, 1].

If you are unable to complete any of the questions in the time allowed writeup a solution and submit it on Thursday.

183

MATHEMATICS 320

First Ungraded Assignment

Dated January 8 1990



lim supn→∞

an = lim infn→∞

an = L.


lim supn→∞

an = lim infn→∞

an = +∞.


an > L + ε




√2, (iii) upper



lim supn→∞


an + lim supn→∞

bn.


lim supn→∞


an + lim supn→∞

bn.

may occur.

184

MATHEMATICS 320

Second [graded] Assignment

Due January 19 1990


lim infn→∞

an+1

an≤ lim inf

n→∞n√


n√


an+1

an.



2. Show thatπ

4≤

∞∑

k=1

1k2 + 1

≤ π

4+ 1.

185

MATHEMATICS 320

Third Assignment

Due January 26 1990


(1

1− x

)2

=∞∑

n=0

(n + 1)xn



n=0

xn

√n + 1

) ( ∞∑n=0

xn

√n + 1

)



n∑

k=1

akbk

∣∣∣∣∣ ≤(

n∑

k=1

(ak)2) 1

2(

n∑

k=1

(bk)2) 1

2

must hold.



n=0

√an

n

also converges.

186

MATHEMATICS 320

Fourth Assignment

Due February 2 1990






limn→∞

fn(xn) = f(x).


(−1)n x2 + n

n2


187

MATHEMATICS 320

Fifth Assignment

Due February 9 1990




2. If∑∞


3. If∑∞

n=1 n|bn| < +∞ and

f(x) =∞∑

n=1

bn sin nx

then

f ′(x) =∞∑

n=1

nbn cosnx

for all x.


∞∑n=0

anxn

1− x=

∞∑n=0

(a0 + a1 + . . . an)xn

is valid.


∞∑n=0

(1 +

11!

+12!

+ . . .1n!

)xn.

188

MATHEMATICS 320

Sixth Assignment

Due February 23 1990


∣∣∣∣∣∫ b

a

f(x)g(x) dx

∣∣∣∣∣ ≤(∫ b

a

f2(x)) dx

) 12

(∫ b

a

g2(x) dx

) 12



|f(x)− f(y)| ≤ M |x− y|α




|an| ≤ C

nkand |bn| ≤ C

nk


189

MATHEMATICS 320

Seventh Assignment

Due February 30 1990


m ≤ σn(x) ≤ M


need not hold.

2. Show thatπ2

8= 1 +

132

+152

+172

+192

. . .

andπ4

96= 1 +

134

+154

+174

+194

. . . .


190

MATHEMATICS 320

Eighth Assignment

Due March 9 1990


0

f(t)tn dt = 0.



1π

∫ π

−π

f(t)g(t) dt = 2a0α0 +∞∑

k=1

anαn + bnβn.


(∫ b

a

(f(x) + g(x))2 dx

) 12

≤(∫ b

a

f2(x)) dx

) 12

+

(∫ b

a

g2(x) dx

) 12


191

MATHEMATICS 320

First Assignment

May 13 1991



lim supn→∞

an = lim infn→∞

an = L.


lim supn→∞

an = lim infn→∞

an = +∞.


an > L + ε




√2, (iii) upper



lim supn→∞


an + lim supn→∞

bn.


lim supn→∞


an + lim supn→∞

bn.

may occur.

192

MATHEMATICS 320

Second Assignment

Monday, May 27, 1991


lim infn→∞

an+1

an≤ lim inf

n→∞n√


n√


an+1

an.



2. Show thatπ

4≤

∞∑

k=1

1k2 + 1

≤ π

4+ 1.

193

MATHEMATICS 320

Third Assignment

Due June 3, 1991


(1

1− x

)2

=∞∑

n=0

(n + 1)xn



n=0

xn

√n + 1

) ( ∞∑n=0

xn

√n + 1

)



n∑

k=1

akbk

∣∣∣∣∣ ≤(

n∑

k=1

(ak)2) 1

2(

n∑

k=1

(bk)2) 1

2

must hold.



n=0

√an

n

also converges.

194

MATHEMATICS 320

Fourth Assignment

Due June 10 1991






limn→∞

fn(xn) = f(x).


(−1)n x2 + n

n2


195

MATHEMATICS 320

Fifth Assignment

Due June 17 1991




2. If∑∞


3. If∑∞

n=1 n|bn| < +∞ and

f(x) =∞∑

n=1

bn sin nx

then

f ′(x) =∞∑

n=1

nbn cosnx

for all x.


∞∑n=0

anxn

1− x=

∞∑n=0

(a0 + a1 + . . . an)xn

is valid.


∞∑n=0

(1 +

11!

+12!

+ . . .1n!

)xn.

196

MATHEMATICS 320

Sixth Assignment

Due June 17 1991


∣∣∣∣∣∫ b

a

f(x)g(x) dx

∣∣∣∣∣ ≤(∫ b

a

f2(x)) dx

) 12

(∫ b

a

g2(x) dx

) 12



|f(x)− f(y)| ≤ M |x− y|α




|an| ≤ C

nkand |bn| ≤ C

nk


197

MATHEMATICS 320

Seventh Assignment

Due July 8 1991


m ≤ σn(x) ≤ M


need not hold.

2. Show thatπ2

8= 1 +

132

+152

+172

+192

. . .

andπ4

96= 1 +

134

+154

+174

+194

. . . .


198

MATHEMATICS 320

Eighth Assignment

Due July 15 1991


0

f(t)tn dt = 0.



1π

∫ π

−π

f(t)g(t) dt = 2a0α0 +∞∑

k=1

anαn + bnβn.


(∫ b

a

(f(x) + g(x))2 dx

) 12

≤(∫ b

a

f2(x)) dx

) 12

+

(∫ b

a

g2(x) dx

) 12


199

MATHEMATICS 320

Final Examination

December 11 1993

Prepare complete articulate responses to six of the following questions. Youmay consult any texts you like, but write up the responses in your own language.You are not to collaborate with any other students. Submit your answers by thenoon on Friday, December 10, 1993.

1. If an is a sequence of positive numbers converging to zero and

limx→1−

∞∑n=1

anxn = c

exists what can you say about∑∞

n=1 an?

2. If an is a sequence of positive numbers tending monotonically to zero, whatcan you say about the series

a1 − 12(a1 + a2) +

13(a1 + a2 + a3)− 1

4(a1 + a2 + a3 + a4) + . . .?

3. If an is a sequence of positive numbers such that∑∞

n=1 an diverges whatcan you say about the series

∞∑n=1

an

1 + an

and ∞∑n=1

an

1 + n2an?

4. Let f be a continuous function on [0, 1] such that∫ 1

0(f(x))2 dx = 0. Show

that f(x) = 0 for all x.


0f(x)xn dx = 0 for all

n = 0, 1, 2, . . . . Show that f(x) = 0 for all x.

6. Using the formula (1 − t)−1 =∑∞

n=0 tn and all necessary justificationsobtain the formula

∫ 1

0

log(1− t)t

dt = −{

112

+122

+132

+142

+152

+ . . .

}.

7. Show how the Fourier series for the function (π − x)/2,

sin x +12

sin 2x +13

sin 3x +14

sin 4x . . .

200

in the interval (0, 2π) can be used to find a sum for the series in question 6.


n=1

un(x)

converges uniformly to a function f(x) on [1,+∞) and suppose that the limit

limx→+∞

un(x) = `n

exists for each integer n. What can you say about the series

∞∑n=1

`n

and the limitlim

x→+∞f(x)?

9. Let f be integrable and odd on [−π, π] and positive on (0, π) with a Fourierseries ∞∑

n=1

bn sin nx.

Show that |bn| < nb1 for all n = 2, 3, 4, . . . .

10. Prove that the numbers

pk = 1 +1

22k+

132k

+1

42k+

152k

. . .

are all rational multiples of π2k. [Hint: consider the Fourier series∑∞

n=1 n−1 sin nx.]

201


December 11 1993

Time: 3 hours

Answer six of the following questions.


limx→1−

∞∑n=1

anxn = c


n=1 an?


a1 − 12(a1 + a2) +

13(a1 + a2 + a3)− 1

4(a1 + a2 + a3 + a4) + . . .?



∞∑n=1

an

1 + an

and ∞∑n=1

an

1 + n2an?


0(f(x))2 dx = 0. Show







∫ 1

0

log(1− t)t

dt = −{

112

+122

+132

+142

+152

+ . . .

}.


sin x +12

sin 2x +13

sin 3x +14

sin 4x . . .

202



n=1

un(x)


limx→+∞

un(x) = `n


∞∑n=1

`n

and the limitlim

x→+∞f(x)?


n=1

bn sin nx.



pk = 1 +1

22k+

132k

+1

42k+

152k

. . .


n=1 n−1 sin nx.]

203

MATHEMATICS 320

Final Examination

December 6 1995


n=1 a2n < ∞. Show that

lim supn→∞

a1 +√

2a2 +√

3a3 +√

4a4 + · · ·+√nan

n< ∞.

[Hint: You may use the Cauchy-Schwartz inequality.]

2. Let {fn} be a sequence of continuous functions on an interval [a, b] suchthat each fn is positive. Suppose that fn converges uniformly to a function fon [a, b] and that f is also positive there.

(i) Show that there is a positive number m so that

fn(x) ≥ m > 0

for all x ∈ [a, b] and all n.

(ii) Show that 1/fn converges uniformly to 1/f on [a, b].

(iii) Suppose that the hypothesis that the functions fn are continuous is dropped.Show that it is not possible to conclude that 1/fn converges uniformly to 1/fon [a, b].

204

3. Let {fn} be a sequence of continuous functions on the interval [0,∞) suchthat for each n

limx→∞

fn(x) = Ln.

Suppose that fn converges uniformly to a function on [0,∞) and that xn is asequence of numbers tending to ∞. Show that

limn→∞

fn(xn) = limn→∞

Ln.

[Hint: First show that {Ln} is a Cauchy sequence.]

4. Let {fn} be a sequence of continuously differentiable functions on theinterval [0, 1] such that f ′n converges uniformly to a function g on [0, 1].

(i) Show that there must exist some sequence of numbers {cn} so that

fn(x) + cn

is converging uniformly on [0, 1].

(ii) Show that the sequence fn(x) need not converge at any x in [0, 1].

5. Compute

limx→∞

∞∑n=1

1n

(1− x

x

)n

.

[Hint: First explain why log(1− t) = t + t2/2 + t3/3 + . . . on [−1, 1).]

6. Let f be continuous on [0,∞) and suppose that the integral∫∞0

f(t) dtconverges7.

Show that the integral ∫ ∞

0

e−xtf(t) dt

converges uniformly on any interval [a, b] (a ≥ 0).

[Hint: Set F (t) =∫∞

tf(s) ds and integrate by parts to obtain

∫ ∞

T

e−xtf(t) dt = e−xT F (T )− x

∫ ∞

T

e−xtF (t) dt.

7. The Fourier series for the function |x| (−π ≤ x ≤ π) is the series

π

2− 4

π

(cosx +

132

cos 3x +152

cos 5x +172

cos 7x . . .

)

7not necessarily absolutely, of course!

205

(i) Explain how the series is obtained.

(ii) Show directly from the series itself that it is converging uniformly on [−π, π].

(iii) To what function must the series be converging? (Explain.)

(iv) Explain how to use the series to find a value for∑∞

1 (2k − 1)−2.

(v) Explain how to use the series to find a value for∑∞

1 (2k − 1)−4.

8. Define the function f on the interval [0, 1] by

f(x) ={

x 0 ≤ x < 1/2x− 1/2 1/2 ≤ x ≤ 1

(i) Let ε > 0. Show that there is a polynomial p(x) so that

∫ 1

0

|f(x)− p(x)| dx < ε.

(i) Is it, moreover, true that for every ε > 0 there is a polynomial p(x) so that

|f(x)− p(x)| < ε?

206

MATHEMATICS 320

Final Examination

August 1991Prepare complete articulate responses to as many of the following questions

as you can. You may consult any texts you like, but write up the responses inyour own language; you are not expected to collaborate with any other students.Submit your answers by the morning of August 6, 1991.


limx→1−

∞∑n=1

anxn = c

exists what (if anything) can you say about∑∞

n=1 an?

2. If an is a sequence of positive numbers tending monotonically to zero, what(if anything) can you say about the series

a1 − 12(a1 + a2) +

13(a1 + a2 + a3)− 1

4(a1 + a2 + a3 + a4) + . . .?


n=1 an diverges what(if anything) can you say about

∞∑n=1

an

1 + an

or ∞∑n=1

an

1 + nan

or ∞∑n=1

an

1 + n2an?


n=1

un(x)


limx→+∞

un(x) = `n

exists for each integer n. What (if anything) can you say about the series∞∑

n=1

`n

207

and the limitlim

x→+∞f(x)?


n=1

bn sin nx.


6. The numberspk = 1 +

122k

+1

32k+

142k

+1

52k. . .

are all rational multiples of π2k. [Hint: consider the Fourier series for∑∞

n=1 n−1 sin nx.]

208

MATHEMATICS 320

Preliminary Final Examination

July 1991

Prepare complete articulate responses to as many of the following questionsas you can. You may consult any texts you like, but write up the responses inyour own language; you are not expected to collaborate with any other students.Submit your answers by 2:30 pm, Monday, July 22.

1. Define what is meant by the Cesaro sum of a series∑∞

n=1 an.

2. Give an example of a series that is summable by this method but is divergentin the ordinary sense.

3. Show that for a convergent series the Cesaro sum of the series is just thesum of the series in the ordinary sense.

4. Show that for a series of nonnegative terms the existence of the Cesaro sumof the series already implies that the series is convergent in the ordinary sense.

5. Is the series1− 2 + 3− 4 + 5− 6 + 7 · · ·

Cesaro summable?

6. Define what is meant by the uniform convergence of a series of functions∑∞n=1 fn(x) on a set E.

7. If∑∞

n=1 fn(x) converges uniformly on an interval [a, b] and g is a continuousfunction on [a, b] then

∞∑n=1

g(x)fn(x)

also converges uniformly on [a, b].

8. If∑∞

n=1 fn(x) converges uniformly on an interval [a, b] then

∞∑n=1

g(x)fn(x)

need not converges uniformly on [a, b] if the hypothesis that g is continuous isdropped.

9. If the series∑∞

n=1 an converges absolutely then

∞∑n=1

anxn

209

converges absolutely and uniformly on [−1, 1].

10. If the series∑∞

n=1 an converges then

∞∑n=1

anxn

converges uniformly on [0, 1].



∫ 1

0

log(1− t)t

dt = −{

112

+122

+132

+142

+152

+ . . .

}. (1)


sin x +12

sin 2x +13

sin 3x +14

sin 4x . . .

in the interval (0, 2π) can be used to find a sum for the series in (1).

210

MATHEMATICS 320

Final Examination

11 April 199012:00–15:00AQ 5007

1.(i) Define what is meant by the Cesaro sum of a series∑∞

n=1 an.

(ii) Give an example of a series that is summable by this method but is diver-gent in the ordinary sense.

(iii) Show that for a convergent series the Cesaro sum of the series is just thesum of the series in the ordinary sense.

(iv) Show that for a series of nonnegative terms the existence of the Cesarosum of the series already implies that the series is convergent in the ordinarysense.

(v) Is the series1− 2 + 3− 4 + 5− 6 + 7 · · ·

Cesaro summable?

2.(i) Define what is meant by the uniform convergence of a series of functions∑∞n=1 fn(x) on a set E.

(ii) If∑∞


∞∑n=1

g(x)fn(x)


(iii) If∑∞


∞∑n=1

g(x)fn(x)


(iv) If the series∑∞


∞∑n=1

anxn


211

(v) If the series∑∞


∞∑n=1

anxn


3.(i) Let f and g be continuous and positive functions on the interval [0, +∞)and such that the integral ∫ ∞

0

f(t) dt

diverges. Show that at least one of the two integrals∫ ∞

0

f(t)g(t) dt or∫ ∞

0

f(t)g(t)

dt

must also diverge.

(ii) Using the formula (1 − t)−1 =∑∞


∫ 1

0

log(1− t)t

dt = −{

112

+122

+132

+142

+152

+ . . .

}.

(iii) Show how the Fourier series for the function (π − x)/2,

sin x +12

sin 2x +13

sin 3x +14

sin 4x . . .

in the interval (0, 2π) can be used to find a sum for the series in (ii).

4. Define what is meant by a function of bounded variation on an interval[a, b].

(i) Show that a curve

x = φ(t), y = ψ(t) (a ≤ t ≤ b)

is rectifiable if and only if the functions φ and ψ have bounded variation on[a, b].

(ii) Let {fk} be a sequence of functions converging pointwise to a function fon an interval [a, b]. If

supk∈N

V (fk, a, b) < +∞

then f too has bounded variation on [a, b].

(iii) Show that it is not enough in (ii) here to assume merely that each functionfk has bounded variation.

212

MATHEMATICS 320

Final Examination Preparation

11 April 1990

1. Define what is meant by the Cesaro sum of a series∑∞

n=1 an. Give anexample of a series that is summable by this method but is divergent inthe ordinary sense.

Show that for a convergent series the Cesaro sum of the series is just thesum of the series in the ordinary sense.

Show that for a series of nonnegative terms the existence of the Cesaro sumof the series already implies that the series is convergent in the ordinarysense.

Is the series1− 2 + 3− 4 + 5− 6 + 7 · · ·

Cesaro summable?

2. If∑∞

n=1 an converges then∑∞

n=1 anxn converges uniformly on [0, 1].

3. Prove that this statement need not be true: if {fn} and {gn} convergeuniformly on a set E then so too does the sequence {fngn}. Can you stateand prove a positive result of this type?

4. If∑∞

n=1 fn(x) converges uniformly on an interval [a, b] and g is a contin-uous function on [a, b] then

∑∞n=1 g(x)fn(x) also converges uniformly on

[a, b].

Can [a, b] be replaced by an unbounded interval? Can g be replaced by adiscontinuous function?

5. Let {fn} be a sequence of continuous functions on an interval [a, b] thatconverges uniformly on the open interval (a, b). Show that the sequencein fact converges uniformly on [a, b].

6. State and give a proof for the Weierstrass M-test for the uniform conver-gence of infinite series.

Give an example of a uniformly convergent series for which the test applies.

Give an example of a series for which the test does not apply even thoughthe series is uniformly convergent.

7. Show that the Fourier series of a 2π–periodic, continuously differentiablefunction converges uniformly.

213


0

f(t)tn dt = 0.

Prove that f vanishes.

9. Let f and g be continuous and positive functions on the interval [0, +∞)and such that the integral ∫ ∞

0

f(t) dt


0


0

f(t)g(t)

dt

must also diverge.

10. Let f be a continuous function on the interval [0, +∞) such that limx→+∞ f(x) =α exists. Determine

limx→+∞

∫ x

0

f(t) logx

tdt.

11. Give all necessary justifications for the computation of the formula

1(y + 1)2

= −∫ 1

0

xy log x dx

from the formula1

(y + 1)=

∫ 1

0

xy dx.


n=0 tn and all necessary justificationsobtain the formulas

∫ 1

0

log(1− t)t

dt = −{

112

+122

+132

+142

+152

+ . . .

}

and, for positive integers p and q,∫ 1

0

tp−1

1 + tqdt =

1p− 1

p + q+

1p + 2q

− 1p + 3q

+1

p + 4+ · · · .

13. Let {fk} be a sequence of functions converging pointwise to a function fon an interval [a, b]. If

supk∈N

V (fk, a, b) < +∞

then f too has bounded variation on [a, b]. Show that it is not enoughhere to assume merely that each function fk has bounded variation.

214

MATHEMATICS 320


February 12 1990


lim supn→∞


an + lim supn→∞

bn.


lim supn→∞


an + lim supn→∞

bn.

may occur.

2. Prove: if {fn} converges uniformly on a set E and g is bounded on E thenthe sequence {gfn} converges uniformly on E. Show that this may not hold ifg is unbounded.

3. Prove: if {fn} is a sequence of continuous functions that converges uniformlyon an interval [a, b] to a function f then f must be continuous too.

4. Let {fn} be a sequence of continuous functions on an interval [a, b] thatconverges uniformly on the open interval (a, b). Show that the sequence in factconverges uniformly on [a, b].


∞∑n=0

anxn

1 + x=

∞∑n=0

(an − an−1 + . . . (−1)na0)xn

is valid.

215

MATHEMATICS 320

Final Examination

11 April 199012:00–15:00AQ 5007

1.(i) Define what is meant by the Cesaro sum of a series∑∞

n=1 an.

(ii) Give an example of a series that is summable by this method but is diver-gent in the ordinary sense.

(iii) Show that for a convergent series the Cesaro sum of the series is just thesum of the series in the ordinary sense.

(iv) Show that for a series of nonnegative terms the existence of the Cesarosum of the series already implies that the series is convergent in the ordinarysense.

(v) Is the series1− 2 + 3− 4 + 5− 6 + 7 · · ·

Cesaro summable?

2.(i) Define what is meant by the uniform convergence of a series of functions∑∞n=1 fn(x) on a set E.

(ii) If∑∞


∞∑n=1

g(x)fn(x)


(iii) If∑∞


∞∑n=1

g(x)fn(x)


(iv) If the series∑∞


∞∑n=1

anxn


216

(v) If the series∑∞


∞∑n=1

anxn


3.(i) Let f and g be continuous and positive functions on the interval [0, +∞)and such that the integral ∫ ∞

0

f(t) dt


0


0

f(t)g(t)

dt

must also diverge.

(ii) Using the formula (1 − t)−1 =∑∞


∫ 1

0

log(1− t)t

dt = −{

112

+122

+132

+142

+152

+ . . .

}.

(iii) Show how the Fourier series for the function (π − x)/2,

sin x +12

sin 2x +13

sin 3x +14

sin 4x . . .

in the interval (0, 2π) can be used to find a sum for the series in (ii).

4. Define what is meant by a function of bounded variation on an interval[a, b].

(i) Show that a curve

x = φ(t), y = ψ(t) (a ≤ t ≤ b)

is rectifiable if and only if the functions φ and ψ have bounded variation on[a, b].

(ii) Let {fk} be a sequence of functions converging pointwise to a function fon an interval [a, b]. If

supk∈N

V (fk, a, b) < +∞

then f too has bounded variation on [a, b].

(iii) Show that it is not enough in (ii) here to assume merely that each functionfk has bounded variation.

217

MATHEMATICS 320

Final Examination

December 11 1993

Prepare complete articulate responses to six of the following questions. Youmay consult any texts you like, but write up the responses in your own language.You are not to collaborate with any other students. Submit your answers by thenoon on Friday, December 10, 1993.


limx→1−

∞∑n=1

anxn = c


n=1 an?


a1 − 12(a1 + a2) +

13(a1 + a2 + a3)− 1

4(a1 + a2 + a3 + a4) + . . .?



∞∑n=1

an

1 + an

and ∞∑n=1

an

1 + n2an?


0(f(x))2 dx = 0. Show







∫ 1

0

log(1− t)t

dt = −{

112

+122

+132

+142

+152

+ . . .

}.


sin x +12

sin 2x +13

sin 3x +14

sin 4x . . .

218



n=1

un(x)


limx→+∞

un(x) = `n


∞∑n=1

`n

and the limitlim

x→+∞f(x)?


n=1

bn sin nx.



pk = 1 +1

22k+

132k

+1

42k+

152k

. . .


n=1 n−1 sin nx.]

219

MATHEMATICS 320

Third Assignment


1. Let {an} be a decreasing sequence of positive numbers. Then∑

an convergesonly if lim nan = 0. Does the converse hold?


Ln =log

(1

an

)

log n.



∑an diverges.


∞∑n=2

xlog n∞∑

n=2

xlog log n∞∑

n=2

(log n)− log n






only if p > 1.

220

MATHEMATICS 320

Final Examination

December 6 1995


n=1 a2n < ∞. Show that

lim supn→∞

a1 +√

2a2 +√

3a3 +√

4a4 + · · ·+√nan

n< ∞.

[Hint: You may use the Cauchy-Schwartz inequality.]

2. Let {fn} be a sequence of continuous functions on an interval [a, b] suchthat each fn is positive. Suppose that fn converges uniformly to a function fon [a, b] and that f is also positive there.

(i) Show that there is a positive number m so that

fn(x) ≥ m > 0

for all x ∈ [a, b] and all n.

(ii) Show that 1/fn converges uniformly to 1/f on [a, b].

(iii) Suppose that the hypothesis that the functions fn are continuous is dropped.Show that it is not possible to conclude that 1/fn converges uniformly to 1/fon [a, b].

221


limx→∞

fn(x) = Ln.

Suppose that fn converges uniformly to a function on [0,∞) and that xn is asequence of numbers tending to ∞. Show that

limn→∞

fn(xn) = limn→∞

Ln.

[Hint: First show that {Ln} is a Cauchy sequence.]

4. Let {fn} be a sequence of continuously differentiable functions on theinterval [0, 1] such that f ′n converges uniformly to a function g on [0, 1].


fn(x) + cn



5. Compute

limx→∞

∞∑n=1

1n

(1− x

x

)n

.

[Hint: First explain why log(1− t) = t + t2/2 + t3/3 + . . . on [−1, 1).]

6. Let f be continuous on [0,∞) and suppose that the integral∫∞0

f(t) dtconverges8.

Show that the integral ∫ ∞

0

e−xtf(t) dt

converges uniformly on any interval [a, b] (a ≥ 0).

[Hint: Set F (t) =∫∞

tf(s) ds and integrate by parts to obtain

∫ ∞

T

e−xtf(t) dt = e−xT F (T )− x

∫ ∞

T

e−xtF (t) dt.

7. The Fourier series for the function |x| (−π ≤ x ≤ π) is the series

π

2− 4

π

(cosx +

132

cos 3x +152

cos 5x +172

cos 7x . . .

)

8not necessarily absolutely, of course!

222


(ii) Show directly from the series itself that it is converging uniformly on [−π, π].



1 (2k − 1)−2.


1 (2k − 1)−4.

8. Define the function f on the interval [0, 1] by

f(x) ={

x 0 ≤ x < 1/2x− 1/2 1/2 ≤ x ≤ 1

(i) Let ε > 0. Show that there is a polynomial p(x) so that

∫ 1

0

|f(x)− p(x)| dx < ε.

(i) Is it, moreover, true that for every ε > 0 there is a polynomial p(x) so that

|f(x)− p(x)| < ε?

223

MATHEMATICS 320

Final Examination

December 4 1997


Ln =log (1/an)

log n.

Show that iflim infn→∞

Ln > 1

then∑

an converges. Show that if Ln ≤ 1 for all sufficiently large n then∑

an

diverges.

2. What assertions about the series∑

an can be made if you are given that

lim supn→∞

log (1/an)log n

> 1?

3. Explain how the test in question #1 can be used to help determine the setof values of x for which the series

∞∑n=2

xlog n

converges. What is the exact set of convergence?

4. What is the best assertion you are able to make about the uniform conver-gence of

∞∑n=2

xlog n?

5. What is the best assertion you are able to make about the value

limx→α+

∞∑n=2

xlog n?


∫ b

a

( ∞∑n=2

xlog n

)dx?

224


d

dx

( ∞∑n=2

xlog n

)?

8. A sequence of functions {fn} defined on an interval I is said to convergecontinuously to the function f if fn(xn) → f(x0) whenever {xn} is a sequenceof points in the interval I that converges to a point x0 in I. Prove the followingtheorem:

Let {fn} be a sequence of continuous functions on an interval [a, b].Then {fn} converges continuously on [a, b] if and only if {fn} con-verges to f uniformly on [a, b].

Does the theorem remain true if the interval [a, b] is replaced with (a, b) or[a,∞)?

You would be expected to complete at least five of these questions in the three hours of the

examination period. If you are unable to complete any of the questions in the time allowed write

up a solution and submit to me or my mailbox by Friday at 4:00pm.

225

MATHEMATICS 320

Final Examination

April 13, 2000

1. Define what it means for a sequence of continuous functions {fn} to convergeuniformly to a function f on an interval I.

2. Show that if {fn} converges uniformly to f on an interval (a, b) and allfunctions f , f1, f2, f3, . . . are continuous on [a, b] then {fn} converges uniformlyto f on [a, b].

3. Show that if {fn} converges uniformly to f on an interval [a, b] and allfunctions f1, f2, f3, . . . are continuous on [a, b] then

limn→∞

∫ b

a

fn(s) ds =∫ b

a

f(s) ds.

4. Show that the conclusion of the preceding problem would not necessarily betrue if the convergence was pointwise and not uniform.

5. Show that

limn→∞

∫ b

a

fn(s) ds =∫ b

a

f(s) ds.

is true under the assumption that all functions f , f1, f2, f3, . . . are continuouson [a, b] and

limn→∞

∫ b

a

(fn(s)− f(s))2 ds = 0.


limx→∞

fn(x) = Ln.

Suppose that fn converges uniformly to a function on [0,∞). Show that

limn→∞

fn(n) = limn→∞

Ln.

7. The Fourier series for the function | sin x| is the series

2π− 4

π

∞∑n=1

cos 2nx

4n2 − 1.

226


(ii) Show directly from the series itself that it is converging uniformly on thewhole real line.



k=11

4k2−1 .


k=11

(4k2−1)2 .

(vi) Establish the identity

| sin x| = 8π

∞∑

k=1

sin2 kx

4k2 − 1.

227

MATHEMATICS 320

First Midterm

February 2, 2001

Prove or disprove each of the the following assertions. You may use any theorem or exampleof the course by citing it briefly. Each of these should respond to a relatively short argument.If you find yourself spending a lot of time you are probably off on a wrong approach.

1. The series∞∑

k=1

k(sinx)k is absolutely convergent for all |x| < π/2.

2. The series∞∑

k=1

ekx

kis convergent for all x < 0 and divergent for all x ≥ 0.

3. If the series∞∑

k=1

ak is absolutely convergent then so too is the series∞∑

k=1

(ak)2.


k=1


k=1

√|ak|.


k=1

βk is absolutely convergent then so too is the series∞∑

k=1

sin βk.

6. If both of the series∞∑

k=1

ak and∞∑

k=1

bk are absolutely convergent then so too

is the series∞∑

k=1

akbk.


k=1

ak and∞∑

k=1


is the series∞∑

k=1

√|akbk|.


k=1

ak converges then so too does the series∞∑

k=1

ak/k.

9. If lim supk→∞

|ktak| < ∞ for some t > 1 then the series∞∑

k=1

ak is absolutely

convergent.

228


|ktak| = ∞ for all t > 1 then the series∞∑

k=1

|ak| is divergent.

229

MATHEMATICS 320

First Midterm

February 2, 2001

Prove or disprove each of the the following assertions. You may use any theorem or exampleof the course by citing it briefly. Each of these should respond to a relatively short argument.If you find yourself spending a lot of time you are probably off on a wrong approach.

1. The series∞∑

k=1

k(sinx)k is absolutely convergent for all |x| < π/2.

Ratio test (or root test) handles this with no troubles.

2. The series∞∑

k=1

ekx

kis convergent for all x < 0 and divergent for all x ≥ 0.

Ratio test (or root test) handles the cases x < 0 and x > 0 immedi-ately. The case x = 0 is just the harmonic series and so divergent.


k=1


k=1

(ak)2.

If the series∞∑

k=1

ak is absolutely convergent then ak → 0 and then

in particular {ak} is bounded, say |ak| ≤ M for all k. Thus we get

that the series∞∑

k=1

(ak)2 converges by comparison with the convergent

series∞∑

k=1

M |ak|


k=1


k=1

√|ak|.

Nope: try ak = 1/k2.


k=1

βk is absolutely convergent then so too is the series∞∑

k=1

sin βk.

230

MATHEMATICS 320

Seventh Assignment

Due July 8 1991


m ≤ σn(x) ≤ M


need not hold.

2. Show thatπ2

8= 1 +

132

+152

+172

+192

. . .

andπ4

96= 1 +

134

+154

+174

+194

. . . .


232

MATHEMATICS 320

Eighth Assignment

Due July 15 1991


0

f(t)tn dt = 0.



1π

∫ π

−π

f(t)g(t) dt = 2a0α0 +∞∑

k=1

anαn + bnβn.


(∫ b

a

(f(x) + g(x))2 dx

) 12

≤(∫ b

a

f2(x)) dx

) 12

+

(∫ b

a

g2(x) dx

) 12


233

MATHEMATICS 320

First Assignment

January 20, 2000



lim supn→∞

an = lim infn→∞

an = L.


lim supn→∞

an = lim infn→∞

an = +∞.


an > L + ε




√2, (iii) upper



lim supn→∞


an + lim supn→∞

bn.


lim supn→∞


an + lim supn→∞

bn.

may occur.

234

MATHEMATICS 320

Second Assignment

January 27, 2000

Warning: you should be attempting many of the exercises in the text. Do noteven consider the weekly assigments as bare minimums, they are just simply allI am willing to take in and read. You cannot learn mathematics at this levelwithout trying lots of different problems involving the concepts.

2. Let {xk} be a sequence of positive numbers. Consider the following condi-tions:(a) lim sup

k→∞

√kak > 0

(b) lim supk→∞

√kak < ∞

(c) lim infk→∞

√kak > 0

(d) lim infk→∞

√kak < ∞

Which condition(s) imply convergence or divergence? Supply proofs.Which conditions are inconclusive as to convergence or divergence? Supply

examples.

235

MATHEMATICS 320

Third Assignment



x0 + (x1 − x0) + (x2 − x2) + (x3 − x2) + . . . .

From this and the identity 1n(n+1) = 1

n − 1n+1 determine the sum of

11 · 2 +

12 · 3 +

13 · 4 +

14 · 5 + . . . .


Ln =log

(1

an

)

log n.



∑an diverges.


∞∑n=2

xlog n,

∞∑n=2

xlog log n,

∞∑n=2

(log n)− log n






only if p > 1.

236

MATHEMATICS 320

Fourth Assignment


1. Form the product of the series∑∞

k=0 akxk with the geometric series

11− x

= 1 + x + x2 + x3 + . . .

and obtain the formula

11− x

∞∑

k=0

akxk =∞∑

k=0

(a0 + a1 + a2 + · · ·+ ak)xk.

For what values of x would this be valid?

2. Verify that ex+y = exey by proving that

∞∑

k=0

(x + y)k

k!=

∞∑

k=0

xk

k!

∞∑

k=0

yk

k!.

237

MATHEMATICS 320

Fifth Assignment


1. Prove: If {fn} and {gn} converge uniformly on a set E then so too does thesequence {fn + gn}.

2. Prove: If {fn} converges uniformly on a set E to a function f and each fn

is bounded on E then so too is f .

3. Prove: If {fn} and {gn} are sequences of bounded functions that convergeuniformly on a set E then so too does the sequence {fngn}.


5. Prove: If {fn} is a sequence of continuous functions converging uniformlyon an interval (a, b) and {xn} is a sequence of numbers converging to a pointx ∈ (a, b) then

limn→∞

fn(xn) = f(x).


(−1)n x2 + n

n2


238

MATHEMATICS 320

Sixth Assignment

Due March 2, 2000

1. Let {fn} be a sequence of continuously differentiable functions on the interval[0, 1] such that f ′n converges uniformly to a function g on [0, 1].


fn(x) + cn



2. Prove or disprove that if f is a continuous function on (−∞,∞) then

f(x + 1/n) → f(x)

uniformly on (−∞,∞). (What extra condition, stronger than continuity, wouldwork if not?)

3. Suppose that fn → f on (−∞,+∞). What conditions would allow you tocompute that

limn→∞

fn(x + 1/n) = f(x)?

Give an example to show that your extra conditions cannot be dropped.

4. Let {fn} be a sequence of continuous functions on an interval [a, b] thatconverges uniformly to a function f . What conditions on g would allow you toconclude that

limn→∞

∫ b

a

fn(t)g(t) dt =∫ b

a

f(t)g(t) dt?

Give an example to show that your extra condition cannot be dropped.

239

MATHEMATICS 320

Seventh Assignment

Due March 17, 2000

1. From the power series

(1 + x2)−1 =∞∑0

(−1)kx2k

give all the necessary justifications (convergence, uniform convergence etc.) toobtain by integration that

π/4 =∞∑0

(−1)k1/(2k + 1).

2. Suppose that you are given that the power series∑∞

k=0 akxk convergesuniformly in (−∞,∞). What can you conclude?

3. Show that if∑∞

k=0 akxk converges uniformly on an interval (−r, r) then itmust in fact converge uniformly on [−r, r].

4. Find a power series expansion about x = 0 for the function

f(x) =∫ 1

0

1− e−sx

sds.

240

MATHEMATICS 320

Eighth Assignment

Due April 1, 2000


m ≤ σn(x) ≤ M


need not hold.

2. Show thatπ2

8= 1 +

132

+152

+172

+192

. . .

andπ4

96= 1 +

134

+154

+174

+194

. . . .



0

f(t)tn dt = 0.



1π

∫ π

−π

f(t)g(t) dt = 2a0α0 +∞∑

k=1

anαn + bnβn.


n=1

bn sin nx.



pk = 1 +1

22k+

132k

+1

42k+

152k

. . .


n=1 n−1 sin nx.]

241

MATHEMATICS 320


February 4, 2000

1. Let {xn} be a sequence of real numbers. Show that

lim supn→∞

x1 + x2 + · · ·+ xn

n≤ lim sup

n→∞xn.

Give an example to show that the two terms may be unequal.[Hint: If for some β and N you have xn < β for all n ≥ N then notice that

x1 + x2 + · · ·+ xn

n≤ x1 + x2 + · · ·+ xN

n+

(n−N + 1)βn

.

Now if you take limsup in this inequality as n → ∞ (keeping N and β fixed)you get some nice information.]

2. Define what it means for a series to be absolutely convergent.(a) Prove that if

∑∞k=1 ak is absolutely convergent then

∑∞k=1(ak)2 is conver-

gent.(b) Is it necessarily true that if

∑∞k=1 ak is nonabsolutely convergent then∑∞

k=1(ak)2 is convergent?

3. Let {xk} be a sequence of nonnegative numbers. Consider the following twoconditions:(a) lim sup

k→∞k2ak < ∞.

(b) lim supk→∞

k2ak > 0.

Which condition(s) imply convergence or divergence? Supply proofs andcounterexamples as needed.

242

MATHEMATICS 320


March 10, 2000

1. Let f , f1, f2, f3, . . . be real-valued functions defined on R and let E ⊂ R.

(a) Prove or disprove: if fn → f uniformly on E then fn → f pointwise on E.

(b) Prove or disprove: if fn → f pointwise on E then fn → f uniformly on E.

(c) Prove or disprove: if fn → f uniformly on E and uniformly on R \ E thenfn → f uniformly on R.

(d) Prove or disprove: if fn → f uniformly on E then fn → f uniformly on E.

(e) Prove or disprove: if fn → f uniformly on E and each fn is continuous thenfn → f uniformly on E.

2. State and prove a result that will allow you to conclude that

limn→∞

∫ b

a

fn(t) dt =∫ b

a

(lim

n→∞fn(t)

)dt.

3. State (without proof) the Weierstrass M-test and use it to discuss the uniformconvergence behaviour of the series

f(x) =∞∑

k=0

ekx.

What can you say about∫ b

af(t) dt? What can you say about f ′(x)?

243

MATHEMATICS 320


March 9, 2001

1. Prove or disprove that∞∑

k=1

cos 2kπx

k

converges uniformly on (−1, 1).


k=1

cos 2kπx

k(k + 1)


3. Justify the following computation or else explain why it is invalid:

∫ 1

0

∞∑

k=0

xk

(k + 1)2dx =

∞∑

k=0

1(k + 1)3

.


lims→ 1

2

∞∑

k=0

(2s

3

)k

=∞∑

k=0

13k

.

5. Let {gk} be a sequence of functions defined on an interval (a, b) such thateach gk is uniformly continuous on (a, b). If gk → g uniformly on the interval(a, b) then prove that g must be also uniformly continuous on (a, b).

6. Let {gk} be a sequence of functions defined on an interval [a, b] such thateach gk is integrable on [a, b]. If gk → g uniformly on the interval [a, b] and g isalso integrable on [a, b] then prove that

limk→∞

∫ b

a

gk(s) ds =∫ b

a

g(s) ds.

244

MATHEMATICS 320


March 9, 2001


k=1

cos 2kπx

k


If this were true then for any ε > 0 there must be (Cauchy criterion)an N so that ∣∣∣∣∣

n∑

k=m

cos 2kπx

k

∣∣∣∣∣ < ε

for all m, n ≥ N and all x in (−1, 1). Take x → 1 from the rightand conclude that ∣∣∣∣∣

n∑

k=m

cos 2kπ

k

∣∣∣∣∣ < ε

all m, n ≥ N . From that it would follow that∑∞

k=11k converges

which is false.


k=1

cos 2kπx

k(k + 1)


This follows immediately from the M-test since∣∣∣∣cos 2kπx

k(k + 1)

∣∣∣∣ ≤1k2

for all x in (−1, 1).

3. Justify the following computation or else explain why it is invalid:∫ 1

0

∞∑

k=0

xk

(k + 1)2dx =

∞∑

k=0

1(k + 1)3

.

This is a power series with an interval of convergence [−1, 1] and sothe term by term integration is indeed valid.

Alternatively use the M-test and conclude uniform convergence on[0, 1].

245


lims→ 1

2

∞∑

k=0

(2s

3

)k

=∞∑

k=0

13k

.

This is a power series with an interval of convergence (−3/2, 3/2)and so it represents a continuous function in that interval, thus theterm-by-term limit is valid.

Alternatively the M-test shows it converges uniformly on say [0, 1]so a limit at 1/2 term-by-term is justified.

5. Let {gk} be a sequence of functions defined on an interval (a, b) such thateach gk is uniformly continuous on (a, b). If gk → g uniformly on the interval(a, b) then prove that g must be also uniformly continuous on (a, b).

Let ε > 0 and select N (by uniform convergence) so that

|fn(t)− f(t)| < ε/2

for all n ≥ N and all t in this interval. Now use uniform continuityof fN to find a δ so that

|fN (x)− fN (y)| < ε/2

for all |x−y| < δ in that interval. Put those two together to get that

|f(x)− f(y)| < ε

for all |x − y| < δ in that interval proving that f is uniformly con-tinuous on (a, b).

6. Let {gk} be a sequence of functions defined on an interval [a, b] such thateach gk is integrable on [a, b]. If gk → g uniformly on the interval [a, b] and g isalso integrable on [a, b] then prove that

limk→∞

∫ b

a

gk(s) ds =∫ b

a

g(s) ds.

Let ε > 0 and select N (by uniform convergence) so that

|gn(t)− g(t)| < ε/(b− a)

for all n ≥ N and all t in this interval. Since all functions here areassumed integrable we can write∣∣∣∣∣∫ b

a

gn(x) dx−∫ b

a

g(x) dx

∣∣∣∣∣ ≤∫ b

a

|gn(x)− g(x)| dx ≤∫ b

a

ε/(b−a) dx ≤ ε

from which the conclusion follows.

246

MATHEMATICS 320


March 10, 2000

1. Let f , f1, f2, f3, . . . be real-valued functions defined on R and let E ⊂ R.

(a) Prove or disprove: if fn → f uniformly on E then fn → f pointwise on E.

(b) Prove or disprove: if fn → f pointwise on E then fn → f uniformly on E.

(c) Prove or disprove: if fn → f uniformly on E and uniformly on R \ E thenfn → f uniformly on R.

(d) Prove or disprove: if fn → f uniformly on E then fn → f uniformly on E.

(e) Prove or disprove: if fn → f uniformly on E and each fn is continuous thenfn → f uniformly on E.

2. State and prove a result that will allow you to conclude that

limn→∞

∫ b

a

fn(t) dt =∫ b

a

(lim

n→∞fn(t)

)dt.

3. State (without proof) the Weierstrass M-test and use it to discuss the uniformconvergence behaviour of the series

f(x) =∞∑

k=0

ekx.

What can you say about∫ b

af(t) dt? What can you say about f ′(x)?

247

MATHEMATICS 320

Seventh Assignment

Due March 23, 2001


m ≤ σn(x) ≤ M


need not hold.


n=1

bn sin nx.


3. The Fourier series for the function | sin x| is the series

2π− 4

π

∞∑n=1

cos 2nx

4n2 − 1.


(ii) Show directly from the series itself that it is converging uniformly on thewhole real line.



k=11

4k2−1 .


k=11

(4k2−1)2 .

(vi) Establish the identity

| sin x| = 8π

∞∑

k=1

sin2 kx

4k2 − 1.

248

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