CHAPTER 1: THE REAL NUMBER SYSTEM1.2 ORDERED FIELD AXIOMS1.2.0. Let a, b, c, d∈R and consider each of the following statements.

a. If a < b and c<d<0, then ac>bd.

TRUE: If a is negative and b is positive or if |a|>|b| and since |c|>|d|, then bd is negative and ac>bd.

FALSE: If |a|<|b| and a is positive then b is positive and ac is a negative number that is less than bd which is a negative number.

d. If a<b-ε for all ε>0, then a<0.

Let a<b-ε for all ε>0.

Case 1: If b < ε

Then obviously a<0.

Case 2: If b>ε

Since no parameters for b exist, a could be anything.

1.2.1 Suppos a,b,c belong to the Reals and a ≤b.

a. Prove a+c≤b+c.

b. If c≥0, prove that axb≤bxc.

For a and b, let * stand for the binary operation in each problem. Assume a, b, c belong to the REALS with a≤b.

By symmetry since ≤b., it follows t hat a*c≤b*c.

1.2.3. The posistive part of an a belong to the Reals is defined by

a+:=[|a|+a]/2 and the negative by a-:=[|a|-a]/2.

a. Prove that a=a++a-.

b. Prove that a+= {a, a≥0 and a-= { 0, a≥0

{0, a≤0 {-a, a≤0

1.2.5 Let a, b belong to the Reals.

a. Prove that if a>2,and b=1+√ (a−1), then 2<b<a.

1.2.7. Let x belong to the Reals.

a. Prove that |x|≤2 implies |x-4|≤4|x-2|.

1.3 THE COMPLETENESS AXIOM1.10 DEFINITION: SUPREMUM. Let E ⊂R.

1. The set E is said to be bounded above there is an M ∈R s.t. a≤M for all a∈E in which case M is the upper bound of E.

2. A number s is called a supremum of the set E s is upperbound of E and s ≤M for all upper bounds M of E. In this case we shall say that E has a finite supremum s and write s=sup E.

Note: By Definition 1.10 a supremum is the smallest upper bound of a E when it exists. To show proof that s=sup E, show that s is an upper bound and s is the smallest upper bound.

1.11 EXAMPLE. If E=[0,1] sup E=1.

By definition of an interval, 1 is an upper bound of E. Let M be any upper bound of E; that is M≥x ∀ x∈ E. Since 1∈E, it follows that M ≥1. Thus 1 is the smallest upper bound of E.

1.12 REMARK. If a set has one upper bound, it has infinitely many upper bounds.

Proof: If M0 is an upper bound for a set E, then so is M for any M> M0.

1.13 REMARK. IF a set has a supemum, then it only has one supremum.

Proof: Let s1 and s2 be suprema of the same set E. Then both s1 and s2 are upper bounds of E, whence by Def 1.10.ii, s1 ≥ s2 and s1 ≤ s2 . We conclude by Trichotomy Property that s1 = s2. Notwithstanding that there is only one supremum in a set.

1.14 THEOREM. [APPROXIMATION PROPERTY FOR SUPREMA. If E has finite supremum and ε>0 is any positive number, then there is a point a∈E such that sup E- ε <a≤sup E.

Proof: Suppose TH. 1.14 is false, then there is an ε0>0 s.t. no element of E lies between s0:=sup E- ε0 and sup E. Since sup E is an upper bound for E, it follows that a ≤s0 ∀a∈E; that is s0 is a upper bound of E. Thus by Def 1.10ii. sup E ≤s0 =sup E- ε0. Adding ε0-sup E to both sides of the inequality we conclude that ε0≤0 is a contradiction.

Note: The Approximation Property can be used to show that the supremum of any subset of intergers is itself an interger.

1.15 THEOREM. IF E⊂Z has a supremum, then sup E∈E. In particular if the supremum of a set, which contains only intergers exists that supremum must be an interger.

Proof: Suppose that s:=sup E and apply the Approximation Property to choose an x0∈E such that s-1<x0≤s. If s=x0, then s∈E, as promised. Otherwise s-1 < x0≤s again to choose x1∈E s.t x0<x1<s.

Subtract x0 from his last inequality to obtain 0<x1- x0<s- x0. Since - x0<1-s, it follows that 0<x1- x0<s+(1-s)=1. Thus x1-x0 ∈Z∩(0,1) a contraditicion by Rem. 1.1.iii. We conclude that s∈E.

POSTULATE 3. [COMPLETENESS AXIOM]. If E is a nonempty subset of R that is bounded avoe, then E has a finite supremum.

1.16 THEOREM [ARCHIMEDIAN PRINCIPLE]. Given real numbers a, b, a>0, there is an interger n∈N s. t . b<na .

Proof: If b<a, set n=1. If a≤b, consider the set E={k∈N : ka≤b } . E is nonempty since 1∈E. Let k∈E (i.e. ka<=b). Since a>0, it follows from the First Multiplicative Property that k≤b }. This proves that E is bounded above by b/a. Thus by the Completeness Axiom and Theorem 1.15, E has a finite supremum s that belongs to E, in particular, s∈N . Set n=s+1. Then n∈N and sincle n is larger than s, n cannot belong to E. Thus na>b.

1.18 THEOREM [DENSITY OF RATIONALS]. If a,b ∈R, there is a q∈Q s.t.a<q<b.

1.19 DEFINITION: INFIMUM. Let Let E ⊂R.

1. The set E is said to be bounded below ther is an m∈R such that a≥m ∀a∈E, in which case m is called a lower bound of theset E.

2. A number t is called an infimum of the set E t is a lower boundof E and t≥m for all lower bounds m of E. In this casw we shall say that E has an infimum t and write t=infE.

3. E is said to be bounded if it is bounded both above and below.

NOTE: SUPREMUM-INFIMUM RELATION, the reflection a set:

-E:={x:x=-a for some a∈E ., i.e. –(1,2]=[-2,-1) So a supremum of a set is the same as the negative of its reflection’s infimum.

1.20 THEOREM. [REFLECTION PRINCIPLE]. Let E⊆R be a nonempty set.

E has a suprememum -E has an infimum, in which case: inf(-E)=-sup(E)

E has an infimum -E has a supremum , in which case: sup(-E)=-inf(E).

1.21 THEOREM. [MONTONE PROPERTY].Suppose that A⊆B are nonempty sets of the R.

1. If B has a supremum, then sup A ≤ sup B.2. If B has an infimum, then inf B ≤ inf A.

1.3 EXERCISES1. Decide which of the following statements is true and which is false.

a. If A and B are nonempty bounded subsets of R, then sup (A∩B)≤sup A.i. Let x ∈ A∩B. Let x=sup(A∩B) let m=sup A then by definition x is

the least upperbound of A∩B and m is the least upper bound of A.

Since there a supremum is unique for a given set, then m is the only least upper bound on an interval in A. x is the least upperbound in A∩B because it’s the least upperbound in the interval that is a part of both A and B. Obviously x<m or x=m. It follows that sup (A∩B)≤sup A.

b. Let c be a positive real number. If A is a nonempty bounded subset of R

and B={cx: x∈ A ¿ ,then(B)=c∗( A) .i. Let c be a positive real number. If A is a nonempty bounded subset

of R and B={cx: x∈ A ¿, then A∈B and sup (A) belongs to B. The operation cx is closed under multiplication. Since c is positive sup(B)=c*sup(A)

c. If A+B:=(a+b: a ∈ A. and b∈B where A and B are nonempty bounded subsets of R, then sup(A+B)=sup(A)+sup(B).

i.d. If A-B=:=(a-b: a ∈ A. and b∈B where A and B are nonempty bounded

subsets of R, then sup(A-B)=sup(A)-sup(B).i. Does a homomorphism exist between supremums under the

operation of addition—NO, now if it were the case that A={c+B or c-B where c∈B, then yes, but this is saying that adding points together means that those points will belong to that set and that is not necessarily true. Therefore c and d are false.

2. Find the infimum and supremum of each of the following sets.a. E={x∈R: x2+2x=3}, then x={1,-3} and sup(E)=1, inf(E)=-3b. E={x∈R: x2-2x+3>x2 and x>0}, then x={-1,3} and sup (E)=3, inf(E)=-1

2. Prove that for each a∈R and each n∈N there exists a rational rn such that |a-rn|<1/n.

a. The inequality|a-rn|<1/n is equivalent to a-1/n <rn<a+1/n. For each n ≥ 1 we can find a rational rn with this property by Theorem 1.18.

3. [Density of the Irrationals]. Prove that if a<b are real numbers, then there is an irrational ε∈R such that a <ε<b. *Refer to analysis intro file

a. Let (a,b), ε∈R and let a <ε<b. Now b-a>0. By the Archimedean property, ∃ a positive interger n s.t. n(b-a)>1 or 1/n<b-a. ∃ an integer m, s.t.

m<na<b+1 or m/n ≤ a ≤ (m+1)/n <y. 4. Prove that a lower bound of a set need not be unique but the infimum of a given

set E is unique.a. If m is a lower bound for a set E, then any m0 with m0 < m is also a lower

bound. If E has two infima s1 and s1, then we have s1 ≤ s2 and s2 ≤ s1, hence s1 = s2.

5. Show that if E is a nonempty bounded subset of Z, then inf E exists and belongs to E.

a. Since E is a nonempty bounded subset of Z, then by definition ∃ an a≥m ∀a∈E s.t. m is the lower bound of E. Since m exists, there is a greatest upper bound of E t.

6. Use the Reflection Principle and analogous results about supremum to prove the following results.

1

a. [Approximation Property for Infima]. Prove that if a set E⊂R has a finite infimium and ε>0 is any positive number, then there is appoint a∈Es.t. inf E+ε>a≥inf E.

b. [Completeness Property For Infima]. If E⊆R is a nonempty set bounded below, then E has a (finite) infimum.

7. Supremum vs. infimuma. Prove that if x is an upper bound of E⊆R and x∈E, then x=sup E.b. Make and prove an analogous statemtn for the infimum of E.c. Show by example that the converse of each of theses statements is false.

8. Suppose that E, A, B ⊂R and E=A∪B. Prove that if E has a supremum and both A and B are nonempty, then sup A and sup B both exists and sup E is one of the numbers sup A or sup B.

9. A dyadic rational is a number of the form k/2n for some k,n∈Z. Prove that if a and b are real numbers and a<b, then there exists a dyadic rational q such that a<q<b.

10.Let xn∈R and suppose that there is an M∈R such that |xn|≤M for each n∈N. Prove that sn=sup {xn,xn+1,…} defines a real number for n∈N and that s1≥s2≥…Prove an analogous result about tn=inf{xn,xn+1,…}.

11. If a,b∈R, b-a>1 about a<k<b.

1.4 MATHEMATICAL INDUCTION1.22 THEOREM. [WELL-ORDERING PRINCIPLE]If E is a nonempty subset of N, then E has a least element (i.e. E has finite infimum and inf E ∈E).

1.23 THEOREM. Suppose for each n ∈N that A(n) is a proposition which satisfies the following two properties:

1. A(1) is true or n=1 is true2. Fore every n ∈N, for which A(n) is true, A(n+1) is also true.

Then A(n) is true for all n ∈N.

1.4EXERCISES0. True or false.

a. If a≥0 and b≠0, then (a+b)n≥bn ∀n∈N .b. If a<0<b, then (a+b)n≤bn ∀n∈N .c. If n∈N is even and both a and b are negative then (a+b)n>an+nan-1b

d. If a≠0, then 1/2n =∑k=0

n(n)(k )

[[(a-2)n-k]/[an2n-k] ∀n∈N .

1. Prove that if x1>2, and xn+1=1+√ xn−1 for n∈N, then 2< xn+1< xn holds ∀n∈N .2. Use the Binomial Formula or the Principle of Induction to prove the following:

a.

1.5 INVERSE FUNCTIONS AND IMAGES1.29 DEFINITION. Let X and Y be sets and f: XY.

1. F is said to be 1-1 (one to one or an injection) x1,x2 ∈ X and f(x1)=f(x2)x1=x2 . each x has to be mapped to some unique y)

2. F is said to be onto (or a surjection ) for each y∈Y there is an x∈X s.t. y=f(x). (every y has to be mapped from some x or more than 1 x).

3. F is called a bijection it is both 1-1 and onto. Sometimes to emphasis, f:XY is 1-1 from X onto Y.

1.30 THEOREM.Let X and Y be sets and f: XY. Then the following three properties are equivalent:

1. f has an inverse.2. f is 1-1 from X onto Y.3. There is a function g: YX such that g(f(x))=x ∀ x∈ X and f(g(y))=y ∀ y∈Y .

Moreover for each f: XY there is only one function g that satisfies 1.30.2 and 1.20.3. It is the inverse function f-1.

1.31 Remark. Let I be an interval and let f: IR. If the derivative of f is either always positive on I or always negative on I, then f is 1-1 on I.

1.33 DEFINITIONLet X and Y be sets and f: XY The image of a set E⊆X under f is the set

f(E):={y∈Y :y=f(x) for some x∈ E}.

The inverse imageof a set E⊆Y under f is the set

f-1 (E):= {x∈ X :f(x)=y for some y∈E}.

1.35 DEFINITIONLet ε={Ea}a∈A be a collection of sets.

The union of the collection ε is the set

∪a∈ A→

Ea:={x:x∈ Ea for some a∈A}

∩a∈ A→

Ea:={x:x∈ Ea ∀a∈A}

1.36 THEOREM. [DEMORGAN’S LAW].Let X be a set and {Ea}a∈A be a collection of subsets of X. If for each E⊆X the cymbol Ec

represents the set X\E, then

1. (∪a∈ A→

Ea)c=(∪a∈ A→

Eac)

2. (∩a∈ A→

Ea)c=(∩a∈ A→

Eac)

1.37 THEOREM.1. Let X and Y be sets and f: XY

2. If Ea is a collection of subsets of X, then f((∪a∈ A→

Ea)=(∪a∈ A→

f(Ea)) and f((∩a∈ A→

Ea)=(∩a∈ A→

f(Ea)).

3. If B and C are subsets of X, then f(C\B)⊇f(C)\f(B).

4. If Ea is a collection of subsets of Y, then f-1((∪a∈ A→

Ea)=(∪a∈ A→

f-1(Ea)) and f-1((

∩a∈ A→

Ea)=(∩a∈ A→

f-1(Ea)).

5. If B and C are subsets of Y then f-1 (C\B)⊇f-1 (C)\f-1 (B).6. If E⊆f(X), then f(f-1(E))=E, but if E⊆X, then f-1(f(E))⊇E

1.5EXERCISES0. Decide which of the following statements are true and which are false.

a. Let f(x)=sinx. Then the function of f: [π2,3 π2

] [-1,1] is a bijection, and its

inverse function is sin-1(x).b. Suppose that A,B, and C are subsets of some set X and that f: XX. If A

∩B≠∅ then f (A)∩ f (B∪C)≠∅ .c. Suppose that A and B are subsets of some set . Then (A\B)c=B\A.d. If f takes [-1,1] onto [-1,1], then f-1(f({0}))=0

1. For each of the following a. prove that f is 1-1 on E and find f(E).

i. f(x)=3x-7, E=Rii. f(x)=e1/x, E=(0, ∞)

iii. f(x)=tanx, E=(π2,3 π2

).

iv. f(x)=x2+2x-5, E=(-∞ ,−6¿b. Find an explicit formula for f-1 on f(E).

2. Find f(E) and f-1€ for each of the following.a. f(x)=2-3x, E=(-1,2)b. f(x)=x2+1, E=(-1,2)c. f(x)=2x-x2, E=[-2,2)

3. Five a simple description of each of the following sets.a. ¿ x∈ [0,1 ] [ x−2 , x+1 ]b. ¿ x∈ [0,1 ] [ x−1 , x+1 ]

c. ¿k∈N [−1k ,1k ]

d. ¿k∈N [−1k ,0 ]e. ¿k∈N [−k ,

1k ]

f. ¿k∈N [ k−1k ,k+1k ]

4. Prove (18)5. Prove Theorem 1.37iii,iv,and v.6. Let f(x)=x2.

a. Find subsets B and C of R s.t. f(C\B)≠f(C)\f(B).b. Find a subset E of R s.t. f-1(f(E))≠E.

7. Let X,Y be sets and f: XY. Prove that the following are equivalent.a. f is 1-1 on X.b. f(A\B)=f(A)\f(B) ∀ subsets A and B of X.c. f-1(f(E))=E for all subsets E of X.d. f(A∩B ¿=f (A)∩f (B)∀ subsets A and B of X.

1.6 COUNTABLE AND UNCOUNTABLE SETS1.38 DEFINITION.Let E be a set.

1) E is said to be finite either E=∅ or ∃a 1-1 function which takes {1,2,3,…n}E for some n∈N .

2) E is said to be countable ∃a 1-1 function that takes NE.3) E is saidt to be most countable E is either finite or countable.4) E is said to be uncountable E is neither finite nor countable.

1.39 REMARK. [Cantor’s Diagonalization Argument].

The open interval (0,1)is uncountable.

1.40 LEMMA. A nonempty set E is at most countable there is a function g: NE.

1.41 THEOREM.1) Suppose A and B are sets

a) If A⊆B and B is at most countable, then A is at most countable.b) If A⊆B and A is uncountable, then B is uncountable.c) R is uncountable.

1.42 THEOREM.

Let Ai={A1,A2,…Ai}.

If the individual elements of Ai is at most countable then the union of those elements is at most countable.

1.43 REMARK. The sets Z and Q are countable but the set of irrationals is uncountable.

1.6EXERCISES0. Decide which of the following statements are true or false.

a. Suppose that E is a set. If there exists a function f: EN, then E is most countable.

i. Let f(E)=N. If this is a bijection, then N is most countable. If this is injective then E could be anything.

b. A dyadic rational is a point x∈R s.t. x=n/2m for some n∈Z∧m∈N . The set of dyadic rationals is uncountable.

c. Suppose that A and B are sets and that f: AB is 1-1. If A is uncountable, then B is uncountable.

d. If E1,E2,… are finite sets, and E:= E1 x E2 x E3…. :={x1,x2…): xj∈Ej ∀ j∈N } then E is countable.

1. Prove that the set of odd integers {1,3,…} is countable.2. Prove that set of rational lattice points in space—that is, the set Q3:={(x,y,z):

x,y,z∈Q} is countable.3. Suppose that A and B are sets and that B is uncountable. If there exists a

function which takes A onto B, prove that A is uncountable.4. Suppose that A is finite and f is 1-1 from A onto B. Prove that Bis finite.5. Let f: AB and g: BC and define g°f: AC by (g°f)(x):=g(f(x)).

a. Show that if f,g are 1-1 (respectively onto), then g°f is 1-1 (respectively onto).

b. Prove that if f is 1-1 from A into B and B0:={y:y=f(x) for some x∈A}, then f-1 is 1-1 from B0 onto A.

c. Suppose that g is 1-1 from B onto C. Prove that f is 1-1 on A (respectively onto B) g°f is 1-1 on A (respectively onto C).

6. Suppose that n∈N and φ : {1,2,3 ,…,n }→ {1,2,3 ,…,n }a. Prove that φ is 1-1 φ is onto.b. [PIGEON HOLE PRINCIPLE]. Suppose that E is a finite set and that f:EE.

Prove that f is 1-1 on E f takes E onto E.7. A number x0 ∈R is called algebraic of degree n if it si the root of a polynomial

P(x)=anxn+…+a1x+a0, where aj∈Z , an≠0 and n is minimal. A number x0 that is not algebraic is called transcendental.

a. Prove that if n∈N∧q∈Q, then nq is algebraic.b. Prove that for each n∈N the collection of algebraic numbers of degree n is

countable.c. Prove that the collection of transcendental numbers is uncountable (the

most famous numbers are π∧e.

CHAPTER 2: SEQUENCES IN R2.1 LIMITS OF SEQUENCESAn infinite seuqnce (more briefly, a sequence) is a function whose domain is N. A

sequence f whose terms are xn:=f(n) will be denoted by x1, x2… or {xn}∞

n=1 or {xn}.

2.1 Definition. A sequence of real numbers {xn} is said to converge to a real number a∈R for every ε>0 there is a N∈N (which generally depends on ε ¿ s . t . n≥N |xn-a|<ε .

EXERCISES 2.10. Decide which of the following statements are true and which are false.

a. If xn converges xn/n also converges.b. If xn does not converge xn/n does not converge.c. If xn converges and yn is bounded, then xnyn converges.d. If xn converges to zero and yn>0 ∀n∈N , then xnyn converges.

1. Using the method of Example 2.2i, prove tthat the following limits exists.a. 2-1/n2 as n∞b. 1+π /√n1 as n∞c. 3(1+1¿n)3 as n∞d. (2n2+1)/(3n2)2/3 as n∞

2. Suppose that xn is a sequence of real numbers that converges to 1 as n∞. Using Definition 2.1, prove that each of the following limits exists.

a. 1+2xn2 as n∞b. (πxn-2)/xnπ-2 as n∞

c. (x2n –e)/xn1-e as n∞

3. For each of the following sequences find two convergent subsequences that have different limtis.

a. 3-(-1)n

b. (-1)3n+2c. (n-(-1)nn-1)/n

4. Suppose that xn∈R.a. Prove that {xn} is bounded there is a C>0 s.t. |xn|≤0 ∀n∈N .b. Suppose that {xn} is bounded. Prove that xn/nk0 as n∞ ∀ k∈N .

5. Let C be a fixed positive constant. If {bn} is a sequence of nonnegative numbers that converges to 0, and {xn} is a real sequence that satisfies |xn-a|≤Cbn for large n, prove that xna.

6. Let a be a fixed real number and define xn:=a ∀n∈N .Prove that the constant sequence xn convergs.

7. Proofsa. Suppose that {xn} and {yn} converge to the same real number. Prove that

xn-yn0 as n∞.b. Prove that the sequence {n} does not converge.c. Show that there exist unbounded sequences xn≠yn which satisfy the

conclusion of part (a).8. Suppose that {xn} is a sequence in R. Prove that xn converges to a EVERY

subsequence of xn also converges to a.

2.2 LIMIT THEOREMS

2.2 EXERCISES0. TRUE OR FALSE.

a.

2.3 BOLZANO-WEIERSTRASS THEOREM

2.4 CAUCHY SEQUENCES

2.5 LIMITS SUPREMUM AND INFIMUM

CHAPTER 3: FUNCTIONS ON RCHAPTER 4: DIFFERENTIABILITY ON RCHAPTER 5: INTEGRABILITY ON RCHAPTER 6: INFINITE SERIES OF REAL NUMBERS

CHAPTER 7: INFINITE SERIES OF FUNCTIONSCHAPTER 8: EUCLIDEAN SPACESCHAPTER 9: CONVERGENCE IN RN

CHAPTER 10: METRIC SPACESCHAPTER 11: DIFFERENTIABILITY ON RN

CHAPTER 12: INTEGRATION ON RN

CHAPTER 13: FUNDAMENTAL THEOREM OF VECTOR CALCULUSCHAPTER 14: FOURIER SERIES