papadimitrakis analysis
TRANSCRIPT
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, - .
i
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- . (-) , Supremum , , . , - , . - . .
, , Supremum, . , - , Bolzano - Weierstrass , , - .
: - . ( ) , , , , , .
.1. . , n n. , .2. , , 10 - . , , .3. Riemann , , Darboux - . , Riemann . Riemann -, Darboux Darboux Riemann.4. . . : R ( ). Rd - . , , () ., .5. , -
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, . , ( , Riemann , - , ) .6. Peano. , , . Foundations of Analysis E. Landau . (-) () Dedekind., , Cauchy -, .7. . , Cauchy - .8. , , , -. : - , . , , .9. , , . , - (, ). , IQ. . , .
, , :
Mathematical Analysis, T. Apostol.Differential and Integral Calculus, R. Courant.The Theory of Functions of Real Variables, L. Graves.Foundations of Analysis, E. Landau.Principles of Mathematical Analysis, W. Rudin.The Theory of Functions, E. C. Titchmarsh.
. , Graves , .
, , ., , , - .
2013.
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1 . 11.1 R R. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Supremum infimum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Supremum. . . . . . . . . . . . . . . . . . . . . 91.4 , , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 . 212.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2 , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.3 . . . . . . . . . . . . . . . . . . . . . . . 312.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612.7 . . . . . . . . . . . . . . . . . . . . 64
3 . 693.1 , . . . . . . . . . . . . . . . . . . 693.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753.3 . . . . . . . . . . . . . . . . . . . . . . 823.4 . . . . . . . . . . . . . . . . . . . . . . . . . 1003.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1023.6 Cauchy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4 . 1074.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1074.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 1154.3 . . . . . . . . . . . . . . . . . . . . . . . . 1204.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1244.5 . . . . . . . . . . . . . . . . . . . . . . . . 1324.6 . . . . . . . . . . . . . . . . . . . . . . . . . 141
5 . 1455.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1455.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1525.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 1625.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1695.5 . . . . . . . . . . . . . . . . . . . . . 1765.6 . . . . . . . . . . . . . . . . . . . . . . . 1915.7 Taylor, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1985.8 , . . . . . . . . . . . . . . . . . . . . . . . . 200
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6 Riemann. 2076.1 Darboux. . . . . . . . . . . . . . . . . . . . . . . . 2076.2 . Darboux. . . . . . . . . . . . . . . . . . . . . . . 2116.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2186.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2206.5 . Riemann. . . . . . . . . . . . . . . . . . . . . . . 235
7 . 2437.1 , . . . . . . . . . . . . . . . . . . . . . . 2437.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2497.3 . . . . . . . . . . . . . . . . . . . . . . . 2577.4 Taylor, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
8 . 2798.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2798.2 - . . . . . . . . . . . . . . . . . . . . . . . . . . . 2858.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2988.4 . . . . . . . . . . . . . . . . . . . . . . . . . . 3078.5 Cauchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3128.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313
9 . 3199.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3199.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3219.3 Weierstrass. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332
10 . 33710.1 . . . . . . . . . . . . . . . . . . . . . . 33710.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34610.3 Taylor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36010.4 . . . . . . . . . . . . . . . . . . . . . . . . . . 366
11 . 37311.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37311.2 , , . . . . . . . . . . . . . . . . . . . . . 37811.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 38911.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39511.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39911.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40311.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416
12 . 42512.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42512.2 - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43212.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43612.4 . . . . . . . . . . 44212.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451
13 . 45713.1 Peano. . . . . . . . . . . . . . . . . . . . . . . 45713.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46313.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46913.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477
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13.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482
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viii
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1
.
1.1 R R.
R. , - , ,N, Z Q. : N = {1, 2, 3, . . . }. , 0.
- R .1
.
1.1. n N, n 2.[] x, y
yn xn = (y x)(yn1 + yn2x+ + yxn2 + xn1). (1.1)
[] nxn1(y x) yn xn nyn1(y x) 0 x y. (1.2)
. [] .[] 0 x y , (1.1) x y, ( ) n yn1 , y x, ( ) n xn1.
BERNOULLI. n N.
(a+ 1)n na+ 1 a 1.
. 2 a 0, y = a + 1 x = 1 (1.2) , 1 a 0, y = 1 x = a+ 1 (1.2).
NEWTON. n N. x, y
(x+ y)n =n
k=0
(nk
)xkynk =
nk=0
n!k!(nk)!x
kynk.
1 11 .2 Bernoulli 1.1.5 5.4.6.
1
-
. 3
(x+ y)n = (x+ y) (x+ y) (n )
, n x y . , k Z 0 k n xkynk x k y n k . , k Z 0 k n, xkynk n k k x. , k n (
nk
)= n!k!(nk)! =
n(n1)(nk+1)k! .
k Z 0 k n (nk
) xkynk.
, , |x| x - x 0. , , |x y| 4 x y. |x y| x y.
, .
1.2. [] a > 0, :
|x| a a x a.
|x| < a a < x < a.
[] x, y :
|x y| |x|+ |y|.
. x y 0 0. , .
. R,
R = R {,+}.
, + , +. :
< x, x < +, < +.
(+) = , () = + .
(+) + x = +, x+ (+) = +, (+) + (+) = +,3 Newton 1.1.5 5.4.8. -
, .4 .
, , d- . , 9 10. 11 .
2
-
() + x = , x+ () = , () + () = .
,
(+) + (), () + (+)
.
(+) x = +, x () = +, (+) () = +,
() x = , x (+) = , () (+) = .
(+) (+), () ()
.
()x = , x() = x > 0,
()x = , x() = x < 0,
()() = +, ()() = .
()0, 0()
.
1+ = 0,
1 = 0.
10
.
x = x > 0,
x = x < 0,
x = 0.
x0 ,
0 ,
,
.,5
|+| = +, | | = +.5 1.8,
2.3.28 2.12.
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R R (a,+), [a,+), (, b), (, b] (,+) {x |x > a}, {x |x a}, {x |x < b}, {x |x b} R, -. R , , (a,+], [a,+], [, b), [, b],[,+), (,+], [,+].
R R . - ( ) ( ) . -, (+)+ (+) = +. , , - . (+) (+) . , ( ) , , . (+)0 , , .
a A , , . , , -, , a (3,+] a +. A [, 2] A .
.
1.1.1. 6 a x b a y b, |xy| ba .
1.1.2. x y < 0 z w < 0, 0 < yw xz.
1.1.3. [] b1, . . . , bn > 0. l a1b1 , . . . ,anbn
u, l a1++anb1++bn u.[]7 1, . . . , n N. l y1, . . . , yn u, l 1y1++nyn1++n u., w1, . . . , wn > 0 w1 + + wn = 1. l y1, . . . , yn u, l w1y1 + + wnyn u.
1.1.4. - x : |x+1| > 2, |x1| < |x+1|, xx+2 >
x+33x+1 ,
(x 2)2 4, |x2 7x| > x2 7x, (x1)(x+4)(x7)(x+5) > 0,(x1)(x3)
(x2)2 0. x x : (, 3], (2,+), (3, 7),(,2) (1, 4) (7,+), [2, 4] [6,+), [1, 4) (4, 8], (,2] [1, 4) [7,+).
1.1.5. Bernoulli Newton .6 , .7 y1, . . . , yn, yk k , -
,
1y1++nyn1++n
= 11++n
y1 + + n1++n yn = w1y1 + + wnyn,
wk = k1++n yk, yk y1, . . . , yn. , w1, . . . , wn > 0 w1 + + wn = 1, w1, . . . , wn w1y1 + + wnyn y1, . . . yn .
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1.2 Supremum infimum.
. - A. A u x u x A. u A. A l l x x A. l A., A , - l u l x u x A.
: u A, u u A , l A, l l A.
1.3. [] l x x > a. l a.[] u x x < b. u b.
. [] ( ) a < l. x =a+l2 a < x < l. , , x > a l x. , , .[] , u < b. x = u+b2 u < x < b., , x < b u x. .
1.2.1. l a [a, b], (a, b], [a, b), (a, b), (a,+), [a,+). l . [a, b], [a, b), [a,+). l , l a, a -. l a . (a, b], (a, b), (a,+). l . a -, l a. . , , l (a,+). , , l x x > a. , , 1.3 l a. (a, b], (a, b), (a,+) l a ., , [a, b], (a, b], [a, b), (a, b), (a,+), [a,+) , a, l a . (, a].
1.2.2. , u b [a, b], [a, b), (a, b], (a, b), (, b), (, b]. , , - . [a, b], (a, b], (, b] 1.3 [a, b), (a, b), (, b). , , , b, u b . [b,+).
1.2.3. (a,+), [a,+), (,+) (, b), (, b], (,+) .
1.2.4. N , 1 , N l 1 . , 1 N N (, 1]. N .
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SUPREMUM. -, .
supremum R. , , .
supremum. R R N. , R N supremum, 12.
, supremum, infimum. : 0 , , .
INFIMUM. -, .
. -, A.
A = {x |x A}.
A A 0. , . A -, A -., l A, l A, A ., , A - , supremum u0 A. u0 A, u0 A. l A u0, l A u0. u0 A. l A u0. u0 A.
. - A infimum A. - A supremum A. infimum supremum A , ,
infA g.l.b.A supA l.u.b.A.
1.2.5. 1.2.1 1.2.2, [a, b],(a, b), (a, b], [a, b) (, b], (, b) supremum, b, [a, b],(a, b), (a, b], [a, b), (a,+), [a,+) infimum, a.
. , , A minimum A. , , A maximum A. minimum maximum A , ,
minA maxA.
[a, b] [a, b) -, - maximum maximum., -, supremum.
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1.2.6. A maximum, supA = maxA, supremum A maximum A., A maxA, maxA A. , maxA A A maxA. maxA A. , A minimum, infA = minA, infimum A minimum A. A = {0} [2, 3] {4} minA = 0 maxA = 4. infA = 0 supA = 4.
1.2.7. minN = 1, infN = 1. , N : n N N n+ 1 N n., N N supremum. N . , , N - . .
1.2.8. A = { 1n |n N} = {1,12 ,
13 ,
14 , . . . } maxA = 1, supA = 1.
, A : 1n A A 1
n+1 A 1n . , A ,
, 0, A infimum. infA .
. - A , infA = . - A , supA = +.
(: ) . A , . , + , , , , , A.
- A supremum infimum. A -, supremum , , supremum +., A , infimum , -, infimum . , - A
infA supA.
, x A infA x x supA , , infA A supA A.
, supA, , +. , infA .
supA infA.
supA (i) (ii):(i) A supA. supA = + , supA , supA A.(ii) supA A., supA = + u, . u A ( A ), x A u < x. + A. supA > 0, . supA A ( supA A), x A supA < x , ( (i)),
supA < x supA.
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x A, supA - . supA A.
, infA (i) (ii):(i) A infA.(ii) infA A.
supremum infimum .
A . A -: x1, x2 A x1 < x2 x x1 < x < x2 x A. : . 1.4 , - R, .
1.4. - A : x1, x2 A x1 < x2 x x1 < x < x2 x A. A .
. u = supA, l = infA,
l u +. , , A [l, u]. x (l, u). x A, x1, x2 A x1 < x < x2. , x A. , (l, u) A. (l, u) A [l, u] :
A = (l, u) A = [l, u] A = (l, u] A = [l, u).
A , , infA supA.
.
1.2.1. max{x, y} = x+y+|xy|2 min{x, y} =x+y|xy|
2 .
1.2.2. (a,+) [a, b) .
1.2.3. l a+ > 0, l a. |a b| > 0, a = b.
1.2.4. a 1 0 < < 1, a 0.
1.2.5. infA = infB supA = supB. A = B;
1.2.6. - A. [infA, supA] R A.
1.2.7. [a, b];
1.2.8. - A. ( ) supA A, : supA = + supA < +. A infA.
1.2.9. - A. supA A A . infA A.
1.2.10. A = [0, 2],A = [0, 2),A = [0, 1]{2}, ,, -, A u = supA : A (u , u] 6= > 0; A (u , u) 6= > 0; , , u / A; l = infA.
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1.2.11. - A u. supA u x u x A. u supA < u x A x > . infA l.
1.2.12. - A,B.[] supA infB x y x A, y B.[] , . A = (, 0], B = [0,+) x y x A, y B. , x y x A, y B. A = (, 0], B = (0,+), A = (4,2), B = (2,+) A = (, 0), B = [1, 13]., , x y x A, y B. ( ) supA, infB x y x A,y B.[] x y x A, y B > 0 x A,y B y x . supA = infB x y x A, y B. ;[] 0 < x y x A, y B > 0 x A, y B yx 1 + . supA = infB x y x A, y B.
1.2.13. - A,B A B = R x < y x A,y B. A,B A B. A = (, ), B = [,+) A = (, ], B = (,+).
1.2.14. - A,B. A supA supB x A < x y B y > .
1.2.15. - A, B A B. infB infA supA supB.
1.2.16. [] - A, B. sup(A B) = max{supA, supB} inf(A B) = min{infA, infB}. sup(A B) min{supA, supB} inf(A B) max{infA, infB}.[] - A A = {x |x A}. sup(A) = infA inf(A) = supA.[] - A,B A+B = {x+ y |x A, y B}. A+B A = [3, 5], B = [1, 7] A = (3, 5), B = (1, 7) ; inf(A+B) = infA+ infB sup(A+B) = supA+ supB.[] - A,B A B = {xy |x A, y B}. A B A = [3, 5],B = [1, 7], A = (3, 5),B = (1, 7) A = (1, 5),B = (2, 7) ; A,B (0,+), inf(A B) = infA infB sup(A B) = supA supB.
1.3 Supremum.
1.2.4 1.2.7.
1.1. N , , supN = +.
9
-
. ( ) N , supN . supN 1 N, n N supN 1 < n. supN < n+ 1 , n+ 1 N.
1.1 :
. l > 0, n N 0 < 1n < l.
, l > 0, 1l , N , n N 1l < n , , 0 0 A. , A 0.
1.5 .
1.5. x k Z k x < k + 1.
. x. 1.1 n N n > x m N m > x. l = m, l, n
l < x < n.
:
k Z, k x, k + 1 x.
k x k = l, k x k Z k l. , , k x k = n., , k Z k x k + 1 > x, k x < k + 1. k k x < k + 1 . k x < k+1 k x < k+1 k, k Z. k < k+1 k < k+1, 1 < k k < 1. k k Z, k k = 0, k = k.
. k Z k x < k+1, 1.5, x [x].
,[x] Z [x] x < [x] + 1.
1.3.2. [3] = 3, [3] = 3, [3.5] = 3, [3.5] = 4.
, , - . , , ( R) Q (R). .
YKNOTHTA . a, b a < b r a < r < b.
10
-
. b a > 0, , , n N 1n < b a. , na+ 1 < nb,
na < [na] + 1 na+ 1 < nb.
r = [na]+1n a < r < b.
.
1.3.1. 1.2.3. l a+ 1n n N, l a. |a b| 1n n N, a = b.
1.3.2. infimum supremum {(1)nn |n N}, N { 1n |n N},{(1)n + 1n |n N}, {
1n |n N} {2
1n |n N},
+n=1[2n 1, 2n],
+n=1[
12n ,
12n1 ].
1.3.3. infimum supremum (a, b) Q = {x Q | a < x < b}.
1.3.4. 1.2.12. A = { 1n |n N}, B = {
1n |n N} x y
x A, y B. x y x A, y B. A = {r Q | r < 0}, B = {r Q | r > 0}.
1.3.5. r a r Q r < b, b a. {r Q | r > a} = {r Q | r > b}, a = b. {r Q | r < a} {r Q | r > b} = , a b. {r Q | r a} {r Q | r b} = Q, b a.
1.4 , , .
. n Z, n 1 (, n N), an :
an = a a (n ).
n Z, n 1 a 6= 0, :
an = 1aa (n ) .
, n = 0 a 6= 0, :a0 = 1.
1.4.3 . . .
., - , ,
. supremum.
1.2. n N, n 2. y 0 x 0 xn = y.
11
-
. 8 n N, n 2 y 0.
X = {x |x 0, xn y}.
, 0 X , X -. , y + 1 1 (y+1)n y+1 > y. x X xn y < (y+1)n , ,x < y + 1. X y + 1. X - , supX .
= supX.
, 0, 0 X . n = y. n < y. y
n
n(+1)n1 > 0 > 0
1 yn
n(+1)n1 . (1.2)
( + )n n n( + )n1(( + )
)= n( + )n1 n( + 1)n1 y n.
( + )n y, + X . , X . n y. n > y.
nynn1 > 0 > 0
ny
nn1 . , (1.2),
n ( )n nn1( ( )
)= nn1 n y.
y ( )n. x X xn y ( )n , , x . X . , X . n y. n y n y n = y., , - xn = y y 0. - xn = y. : 1, 2 0, 1n = y 2n = y, 1n = 2n, 1 = 2.
. n N, n 2. y 0, - xn = y, 1.2, n- y
ny.
n = 2, 2y y.
ny - y. , n- ( - xn = y) n
0 = 0
ny > 0 y > 0., - xn = y
y 0, . n , xn = y (i) , ny ny, y > 0, (ii) , 0, y = 0, (iii) , y < 0. n , xn = y (i) , ny, y > 0, (ii) , 0, y = 0, (iii) , n
y, y < 0. .
. ny. , supremum. .
8, 4.4.20 4 n- ny - y. 2.4.12.
12
-
1.4.4 , , . .
1.2 . , n N, xn (, , --) [0,+) [0,+). . 1.2 xn [0,+) [0,+). , y = xn , [0,+), , [0,+)., , x = ny [0,+) [0,+). , y = xn ( [0,+)) x = ny . , y = xn , x = ny .
- - . .
1.6. n, k. nk k n-
. , nk , .
. k n- , m N k = mn. n
k = m , , .
, nk , n
k = ml , m, l N. -
, , m, l > 1. n
k = ml l
nk = mn. , > 1 , . , l lnk, mn = mn1m. l - > 1 m, l mn1 = mn2m. , l mn2 = mn3m. , l m0 = 1. l = 1, k = mn k n- .
, , R .
1.7. R \Q .
. , , 2 , 1.6
2 .
1.2 , , supremum.
, , - . ( R) R \Q ( R).
YKNOTHTA . a, b a < b x a < x < b.
. a < b. c, c =
2. a c < b c,
r a c < r < b c. r + c a < r + c < b.
- . -
- .
1.1. y > 0,m, k Z, n, l N, n, l 2 mn =kl . ( n
y)m = ( l
y)k.
13
-
. c = ( ny)m d = ( ly)k, c = d., ,
cnk = ( ny)mnk =
(( ny)n
)mk= ymk, dml = ( l
y)kml =
(( ly)l
)mk= ymk
, ,cnk = dml. (1.3)
, nk = ml. nk = ml 6= 0, (1.3) c = d. nk = ml = 0 , , m = k = 0, ( ny)m = ( ly)k = 1 .
. , y > 0 r Q r . m Z, n N n 2 r = mn . , 1.1, ( n
y)m
. ,
yr = ( ny)m,
m Z, n N n 2 r = mn ., r Q r > 0,
0r = 0.
n N n 2 y1/n = ny y 0. .
, yr > 0 y > 0 r. 1.4.5 .
, , . .
, yr - r - y , , r 0, yr y.
., x y > 1.
X = {yr | r Q, r < x}.
X - r < x ( , r = [x]) yr X . X . , r Q r < x r < [x] + 1 , , yr < y[x]+1( ). y[x]+1 X . supX .
. 9 y > 1 x ,
yx = sup{yr | r Q, r < x}.
, y = 1 x ,
1x = 1.
9 2.4.20.
14
-
0 < y < 1 x , 1y > 1, (1y )
x
yx = 1(1/y)x .
, x x > 0,
0x = 0.
, , . y < 0, yx x .10 y = 0, yx x . y > 0, yx x.
1.2. y > 1. b > 1 n N 1 < y1/n < b.
. y > 1 b > 1, , b1y1 > 0, n N 0 < 1n y 1
, , y1/n < b. 1 < y1/n .
1.3. x Q y > 1. yx = sup{yr | r Q, r < x}.
. x Q y > 1. r Q r < x, yr < yx. yx {yr | r Q, r < x}. a {yr | r Q, r < x}. , a > 0. ( ) a < yx, y
x
a > 1. , 1.2, n N 1 < y1/n < y
x
a , , a < yx(1/n). , ,
x 1n Q x1n < x , , y
x(1/n) {yr | r Q, r < x}. yx a. yx {yr | r Q, r < x}.
1.3 yx = sup{yr | r Q, r < x} x . , x : yx.
1.4. [] r Q r < x1 + x2. r1, r2 Q r1 < x1 r2 < x2 r1 + r2 = r.[] x1, x2 > 0, r Q 0 < r < x1x2. r1, r2 Q 0 < r1 < x1 0 < r2 < x2 r1r2 = r.
. [] r Q r < x1 + x2. r x1 < x2, r2 Q r x1 < r2 < x2. r1 = r r2, r1 Q. , r1 < x1 r2 < x2 r1 + r2 = r.[] x1, x2 > 0, r Q 0 < r < x1x2. 0 < rx1 < x2, r2 Q 0 0. y1xy2x = (y1y2)x, yx1yx2 = yx1+x2 , (yx1)x2 =yx1x2 .[] x1 < x2. y > 1, yx1 < yx2 . 0 < y < 1, yx1 > yx2 .[] 0 < y1 < y2. x > 0, y1x < y2x. x < 0, y1x > y2x.
. [] .(i) y > 1, yr yx r Q r < x.( yx {yr | r Q, r < x}.)(ii) y > 1 yr a r Q r < x, yx a.(, , a {yr | r Q, r < x} yx .) . y1, y2 > 1. r Q r < x. (y1y2)r = y1ry2r y1xy2x. (y1y2)x y1xy2x. r1, r2 Q r1, r2 < x. r = max{r1, r2} Q, r < x, y1r1y2r2 y1ry2r = (y1y2)r (y1y2)x. y1r1 (y1y2)
x
y2r2. y1x (y1y2)
x
y2r2.
y2r2 (y1y2)x
y1x, , y2x (y1y2)
x
y1x. y1xy2x (y1y2)x.
(y1y2)x y1xy2x y1xy2x (y1y2)x y1xy2x = (y1y2)x. . . y > 1. r1, r2 Q r1 < x1 r2 < x2. yr1yr2 = yr1+r2 yx1+x2, , yr1 y
x1+x2
yr2 . yx1 y
x1+x2
yr2 . yr2 y
x1+x2
yx1 , ,
yx2 yx1+x2
yx1 . yx1yx2 yx1+x2 .
r Q r < x1 + x2. r1, r2 Q r1 < x1 r2 < x2 r1 + r2 = r. yr = yr1+r2 = yr1yr2 yx1yx2 . yx1+x2 yx1yx2 . yx1yx2 yx1+x2 yx1+x2 yx1yx2 yx1yx2 = yx1+x2 . . . y > 1 x1, x2 > 0. r1, r2 Q r1 < x1 r2 < x2. r1, r2 Q r1 r1 r2 r2 0 < r1 < x1 0 < r2 < x2. (yr
1)r
2 = yr
1r
2 yx1x2 , ,
yr1 yr1 (yx1x2)1/r2 . yx1 (yx1x2)1/r2 . yx1 yr1 > 1, (yx1)r2 (yx1)r
2 yx1x2 , , (yx1)x2 yx1x2 .
r Q r < x1x2. r Q r r 0 < r < x1x2. r1, r2 Q 0 < r1 < x1 0 < r2 < x2 r1r2 = r. yr yr
= yr1r2 =
(yr1)r2 (yx1)r2 (yx1)x2 . yx1x2 (yx1)x2 . (yx1)x2 yx1x2 yx1x2 (yx1)x2 (yx1)x2 = yx1x2 . .[] x1 < x2 y > 1. r1, r2 Q x1 < r1 < r2 < x2. r Q r < x1 yr yr1 . yx1 yr1 . , yr1 < yr2 yx2 . yx1 < yx2 . 0 < y < 1 .[] 0 < y1 < y2 x > 0. 1 < y2y1 , [], 1 = (
y2y1)0 < (y2y1 )
x , [], y1x < y1x(y2y1 )x = (y1
y2y1)x = y2
x. x < 0 .
. . y > 1, yx x R
16
-
, 0 < y < 1, yx x R. , . x > 0, yx -
y [0,+) , x < 0, yx y (0,+).
.
a+ = + a > 1, a+ = 0 0 a < 1,
a = 0 a > 1, a = + 0 < a < 1,
(+)b = + b > 0 b = +, (+)b = 0 b < 0 b = .
00, 1+, 1, (+)0, 0
.
1.1. , , , .
., , .
1.3. a > 0, a 6= 1. y > 0 x ax = y.
. , a > 1 y > 1.
X = {x | ax y}.
, 0 X . , 1.2, n N an > y. , x X ax y an, x n. X - , supX .
= supX.
a = y. a < y. 1.2 n N a1/n < y
a. a+(1/n) < y,
+ 1n X . X . a y.
a > y. 1.2, , n N a1/n < ay . y < a(1/n),
x X ax y < a(1/n) , , x < 1n . 1n
X . X . a y. a y a y a = y. . a > 1 y = 1, ax = y x = 0. a > 1 0 < y < 1, , 1y > 1, a
= 1y , , = a = a = y., 0 < a < 1 y > 0, , 1a > 1, (
1a)
= y, = a = a = y. ax = y : a1 = y a2 = y, a1 = a2 , 1 = 2.
17
-
. 11 y > 0 a > 0, a 6= 1, ax = y, - 1.3, y a
loga y.
a > 1. , , y = ax (,+) (,+). y = ax ., 1.3 (0,+). , y = ax (,+) (0,+). , - , x = loga y (0,+) (,+). y = ax x (,+), x = loga y y (0,+).
0 < a < 1, a > 1 . y = ax x (,+) (0,+) x = loga y y (0,+) (,+). , .
1.9 .
1.9. a, b > 0, a, b 6= 1.[] loga(y1y2) = loga y1 + loga y2 y1, y2 > 0.[] loga(yz) = z loga y y > 0 z.[] logb y =
loga yloga b
y > 0.[] loga 1 = 0, loga a = 1.[] 0 < y1 < y2. a > 1, loga y1 < loga y2. 0 < a < 1, loga y1 > loga y2.
. [] x1 = loga y1 x2 = loga y2, ax1 = y1 ax2 = y2. ax1+x2 = ax1ax2 = y1y2, loga(y1y2) = x1 + x2 = loga y1 + loga y2.[] x = loga y, ax = y. azx = (ax)z = yz , , loga(yz) = zx =z loga y.[] x = logb y w = loga b, bx = y aw = b. awx = (aw)x = bx = y. loga y = wx = loga b logb y.[] loga 1 = 0 a0 = 1 loga a = 1 a1 = a.[] x1 = loga y1 x2 = loga y2, y1 = ax1 y2 = ax2 . ax1 < ax2 , a > 1, x1 < x2 , 0 < a < 1, x1 > x2.
.
1.4.1. infimum supremum (a, b) (R \Q) = {x R \Q | a < x < b}.
1.4.2. a A = {r Q | r < a}. A Q A - Q. , u Q (, u = [a] + 1) r u r A. Q, A. Q supremum.
1.4.3. [] a, b > 0 m,n Z, anbn = (ab)n, aman = am+n, (am)n = amn.[] m,n Z m < n. a > 1, am < an. 0 < a < 1, an < am.[] 0 < a < b n Z. n > 0, 0 < an < bn. n < 0, 0 < bn < an.
11 2.4.21.
18
-
1.4.4. n,m N n,m 2.[] y, y1, y2 0, n
y1y2 = n
y1 n
y2 n
my = nm
y.
[] n < m. y > 1, my < ny. 0 < y < 1, ny < my.[] 0 y1 < y2. n
y1 < n
y2.
1.4.5. r, r1, r2 Q.[] y, y1, y2 > 0, y1ry2r = (y1y2)r, yr1yr2 = yr1+r2 , (yr1)r2 = yr1r2 .[] r1 < r2. y > 1, yr1 < yr2 . 0 < y < 1, yr1 > yr2 .[] 0 < y1 < y2. r > 0, y1r < y2r. r < 0, y1r > y2r.
1.4.6. y > 1. yx = inf{yr | r Q, x < r}.
1.4.7. 3, 7
129 3
2 +
5 .
1.4.8. 12 [] n, k,m N k,m > 1. nk/m k,m n- .
[] a0, a1, . . . , an Z k,m Z > 1. km anxn + an1xn1 + + a0, k a0 m an.
1.4.9. m,n N, m,n 2. , m,n, logm n .
12 1.6.
19
-
20
-
2
.
2.1 .
. ( ) x : N R N . n N x(n) , , xn. , n N
xn = x(n).
, , - : x1, x2, x3 . , / -. xn+1 xn xn1 xn. n, N, - . x : N R ,,
(x1, x2, . . . , xn, . . . ) (xn) (xn)+n=1.
, , , x, n, : (yn), (xk), (zm) .
, , . , n,m, k, , - ( 0), n N m N k N.
2.1.1. ( 1n) (1,12 ,
13 , . . . ,
1n , . . . ).
2.1.2. (n) (1, 2, 3, 4, . . . , n, . . . ).
2.1.3. (1) (1, 1, 1, . . . , 1, . . . ).
2.1.4. ((1)n1) (1,1, 1,1, . . . , 1,1, . . . ).
2.1.5. (
110n
) ( 110 ,
1102, 1103, . . . , 110n , . . . ).
2.1.6. n- n, (1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, . . . ). n- .
2.1.7. (m n)+n=1 (m 1,m 2,m 3, . . . ,m n, . . . ).
2.1.8. (m n)+m=1 (1 n, 2 n, 3 n, . . . ,m n, . . . ).
21
-
- (xn)+n=1, (xn), , , .
: . -, . (1)+n=1 {1} . , , (1, 1, 1, . . . ). , ., . , (1,1, 1,1, 1,1, . . . ), (1, 1,1, 1, 1,1, 1, 1,1, . . . ) {1, 1}. (1, 12 ,
13 ,
14 ,
15 ,
16 , . . . ), (
12 , 1,
14 ,
13 ,
16 ,
15 , . . . )
{ 1n |n N}.
. (xn) xn+1 xn n, - xn+1 > xn n, xn+1 xn n xn+1 < xn n. (xn) . (xn) , c xn = c n. , (c) (c, c, c, . . . , c, . . . ).
. 2.1.1, 2.1.5, 2.1.7 , 2.1.2, 2.1.8 , 2.1.3 2.1.4, 2.1.6 .
. (xn) , u : xn u n. u (xn). (xn) -, l : l xn n. l (xn). (xn) , - l u : l xn u n.
, u (xn), u u , l (xn), l l .
2.1.9. (c) .
2.1.10. ( 1n),( (1)n1
n
), (n1n ), ((1)
n1) [1, 1].
2.1.11. ( (1+(1)n1)n
2
) (1, 0, 3, 0, 5, 0, 7, 0, . . . ) -
. , l 0 . , , . .
2.1.12. (1, 0,3, 0,5, 0,7, 0, . . . ), -, . l , l, l, .
22
-
2.1.13. ((1)n1n) (1,2, 3,4, 5,6, . . . ) - . , , - , , .
(xn) , l, u
l xn u
n. , xn [l, u] n. , , 0 [M,M ] [l, u],
M xn M
, , |xn| M n. , (xn) , M |xn| M n. : M |xn| M n, M xn M n, M M (xn). : (xn) M |xn| M n.
n. : n 234 4 n n2 n > 8 xn < xn+1 (xn).
. , n, , , n n0 n n0.
n (xn) (xn), (xn) , , n . : xn+1 xn, xn+1 > xn,xn u, xn = c. , (xn) , , , , u, c.
2.1.14. (1, 23 , 7,2,1,1,1,1, . . . ) , .
2.1.15. (n2 14n + 8) . , n2 14n+8 = (n 7)2 41 n 7 , , .
n n0, n0 n n0 n n0.
n0 = max{n0, n0}.
n0 n0, n n0. , n0 n0, n n0. n n0. : , , , .
2.1.16. n2 3n 37 n 8. , 2n+1n+1 >2513
n 13. n2 3n 37 2n+1n+1 >2513 n max{8, 13} = 13.
23
-
, , - : , n0, n0, , . ,, !
. n .
2.1.17. (1)n1 > 0 n, n. , (1)n1 0 n, n.
n n . , n , , , n.
.
2.1.1. , .
2.1.2. (a+b2 + (1)
n1 ab2
).
m N, (n m[ nm ]
)+n=1
. m = 1, 2, 3.
2.1.3. : 1, 4, 9, 16, 25, 36. 49; 24; ;
2.1.4. - .
2.1.5. (xn) (yn) (xn + yn). , , . , , . (xn) (yn) (xnyn). , , , xn, yn 0 n.
2.1.6. ((1)n1n),( (1)n1
n
),(
18nn2+n+1
),(13n
n!
),(n30
2n
),(2[n2 ]
),(n 3[n3 ]
)
; ; ;
2.1.7. n k n > k . n .
2.1.8. , , , , . , - .
2.1.9. (xn) . c xn = c n.
2.1.10. 1 a, b, p, q, p, q 0, (xn) x1 = a x2 = b xn+2 = pxn+1 + qxn.
1 n- .
24
-
(i) p 6= 0, q = 0. xn = bpn2 n 2.(ii) p = 0, q 6= 0. xn = aq
n12 n xn = bq
n22
n.(iii) p 6= 0, q 6= 0. x2 = px + q , 1, 2, , xn = 1
n1 + 2n1 n.
x2 = px + q , , , xn =
n1 + (n 1)n1 n. x2 = px+ q , , , [0, 2) xn = n1 cos(n 1) + n1 sin(n 1) n. n- 2 x1 = x2 = 1 xn+2 = 3xn, xn+2 = xn+1 + xn, xn+2 =2xn+1 xn, xn+2 = xn+1 xn.
2.1.11. ( ) (xn) xn+1 = x1 + + xn ; xn+3 = xnxn+2xn+1 , xn+1 = 1
1xn
xn+1 = 2 1xn .
2.2 , .
2.2.1. ( 1n) :
1, 12 ,13 ,
14 ,
15 , . . . ,
1100 ,
1101 , . . . ,
1100000 , . . . ,
1100000000 , . . . .
, n , 1n , , 1n 0.
2.2.2. (n1n ) :
0, 12 ,23 ,
34 ,
45 , . . . ,
99100 ,
100101 , . . . ,
99999100000 , . . . ,
99999999100000000 , . . . .
, n , n1n 1 . , n1n 1 |n1n 1| =
1n , , n ,
1n .
(xn) : n , xn x .. xn x |xnx| . n - n . ,, : |xnx| n .
:
2.2.1. ( 1n). 1n 0 |
1n 0| =
1n , , -
n .2 Fibonnaci 1, 1, 2, 3, 5, 8, 13.
25
-
; , 0.000132. 1n 0.000132 n - ; , : n 1n 0.000132; : 1n < 0.000132 n >
1000000132 = 7575.75 . . . .
n > 1000000132 ; 7576 >1000000
132 , , n - 7576, 1n < 0.000132. , 0.0000000000132. , n - 75757575758, 1n < 0.0000000000132. , , : ( 0.000132 0.0000000000132) ( 7576 75757575758). , , . - . , , . , , , n0, , n n0, 1n < . , , - : > 0 n0 1n0 ,, 1n0 ,
1n0+1
, 1n0+2 , . . . < . ( ) n0 . n0 ( ) 1n < ; , . 1n < n >
1 (, n)
.
2.1. a 0, n0 = [a] + 1 n > a. a < 0, n0 = 1 n > a.
. .
: n > 3 1, n > 83 3 = [83 ] + 1 n > 2 3 = 2 + 1 = [2] + 1. ( 1 0) n0 n0 = [
1 ] + 1.
, ( ) n0 , n0.
, .
. (xn) x (xn) x x (xn) > 0 n0 |xn x| < n n0.: (xn) x > 0 |xn x| < . (xn) x
xn x limxn = x limn+ xn = x.
(xn) , (xn) .
, , (xn) x n- xn x n .
:
xn x [ > 0 n0 N n N (n n0 |xn x| < )
]26
-
xn x > 0 n0 ( )
n n0 () |xn x| <
, ,
|xn x| < () n n0.
|xn x| < n n0. : n . :
|xn x| < P1 Pn n n0,
P1, . . . , Pn . : - . , -. . .
2.2.1 .
2.2.1.
1n 0.
> 0. n0 | 1n 0| < n n0. , , | 1n 0| < n n0., | 1n 0| < ( )
1n <
( ) n > 1 . :
| 1n 0| < 1n < n >
1 .
, 1.1 n0 n0 > 1 . , - n > 1 n0 n n0. n0 n n0 n > 1 |
1n 0| < . :
n n0 n > 1 1n < |
1n 0| < .
( ) n0. , 1 0, n0 = [1 ] + 1 n >
1 . , n0
n0, | 1n 0| < n n0.
2.2.3. (c) c.
c c.
> 0. n0 |c c| < n n0. |cc| < ( ) 0 < . , 0 < n ( n). , n0 ( , n0 = 1), |c c| < n n0.
2.2.4. ((1)n1) . ( ) ((1)n1) x. > 0 n0 n n0 |(1)n1x| < . ,
27
-
n0, n n0., n n0 |1x| < n n0 |1 x| < . | 1 x| < 1 (x , x+ ) |1x| < 1 (x , x+ ). , > 0 1, 1 (x , x+ ). , , 0 < 1, 2, .
2.2.5. (1)n1
n 0.
> 0. (1)n1
n 0 < ( ) 1n <
( ) n > 1 . , n0 >1 .
n0, n n0 n > 1 |1n 0| < .
2.2.6. n2+12n3n 0.
> 0. n2+12n3n 0
< ( ) n2+12n3n <
( ) 2n3 n > 1 (n2 + 1). ,
n, . ( )
n2+12n3n
n2+n2
2n3n3 =2n ,
n2+1
2n3n < (, ) 2n < -
( ) n > 2 . . n0 > 2 . n0, n n0 n >
2
n2+12n3n 0
< . n
2+12n3n , ,
2n . -
, , :
a b, a < b < .
a < b < . , b a a.
. (xn) + (xn) + + (xn) M > 0 n0 xn > M n n0. : (xn) + M > 0 xn > M . (xn) +,
xn + limxn = + limn+ xn = +.
, (xn) (xn) (xn) M > 0 n0 xn < M n n0. : (xn) M > 0 xn < M . (xn) ,
xn limxn = limn+ xn = .
, , : (xn) + n- xn + n . .
:
xn + [M > 0 n0 N n N (n n0 xn > M)
]xn
[M > 0 n0 N n N (n n0 xn < M)
]28
-
2.2.7.
n +.
M > 0. n0 n > M n n0. . 1.1, n0 > M . , n > M n0 n n0. n0 n n0 n > M . ( ) n0 = [M ] + 1 , n0 n0, n > M n n0.
2.2.8. ((1)n1n) (1,2, 3,4, 5,6, . . . ) + . ( ) +. M > 0 n0 (1)n1n > M n n0. , n0, n n0. n n0 n > M n n0 n > M . , n > M , , -, . , .
2.2.9. n2+nn+3 +.
M > 0. n3+n
n2+3> M n3+n > Mn2+3M .
n, n3+n
n2+3> M n
3+nn2+3
. ( )
n3+nn2+3
n3n2+3n2
= n4 .
n3+n
n2+3> M n4 > M n > 4M . -
n0 n0 > 4M , , n0, n n0 n > 4M n
3+nn2+3
> M . n
3+nn2+3
, , n4 . , :
a b, a > M b > M.
a > M b > M . - , b a a.3
, lim, limn+ , . - , . limn+ xn, , R.
, .
. > 0. (x , x+ ) - x. (1 ,+] - + [,
1 ) - .
Nx() = (x , x+ ), N+() = (1 ,+], N() = [,1 ).
3 2.2.6.
29
-
M = 1 , =1M
+ (1 ,+] (M,+] [,1 ) [,M).
: x R , Nx() . x Nx(). Nx() x Nx()., . l < x, x ( +) l, > 0 Nx() l., x < u, x ( ) u, > 0 Nx() u., , l < x < u, x () l, u, > 0 Nx() l, u.
|xn x| < xn x , ,x < xn < x+ , , xn (x , x+ ) , , xn Nx()., xn > M xn < M xn , , xn (M,+] xn [,M) , , xn N+() xn N(), = 1M .
4
.
. x R. xn x > 0 n0 xn Nx() n n0 , , > 0 xn Nx().
, , - . , , , - .5
.
2.2.1. , : 1n+8 0,3n+12n+5
32 ,
1n+5
0,n2 7n +, n
2n21 0,n4n2+1n2+3
+, 2n 2n/2 +, 3+log2 n1+3 log2 n 13 .
2.2.2. .
2.2.3. x, y R, x 6= y. > 0 Nx() Ny() = .
2.2.4. x R. , 0 < 1 2, Nx(1) Nx(2). > 0 n Nx( 1n) Nx().
>0Nx() = {x} , ,
x x.
+n=1Nx(
1n) = {x}.
2.2.5. xn x. > 0 n0() n0 xn Nx() n n0. , 0 < < , n0() n0().
4 , xn + xn N+() = ( 1 ,+] xn N() = [,
1).
5 , .
30
-
2.2.6. 6 (xn) x > 0 xn / Nx() n.
2.2.7. 7 0 > 0. xn x 0 < 0 xn Nx().
2.2.8. (xn) x : n0 > 0 |xn x| < n n0. xn x. (xn) .
2.2.9. [] (xn) . , xn x, (xn) x .[] (xn) xn Z n. xn x, (xn) x Z.
2.3 .
2.1 , , .
2.1. . , .
. (xn), (yn) k0,m0
xk0 = ym0 , xk0+1 = ym0+1, xk0+2 = ym0+2, . . . . (2.1)
xn a R. > 0. xn Na() . (2.1) yn Na() . , xn Na() n0 . . . n0 k0, xn Na() k0 , , yn Na() m0 . . n0 > k0, n0 = k0 + p yn Na() m0 + p . yn a.
2.3.1. (1, 12 ,13 ,
14 ,
15 ,
16 , . . . ), (2, 5,
14 ,
15 ,
16 , . . . ). -
0. , - . 0.
2.3.2. (xn) (x1, x2, x3, . . . ). (xn+1) (x2, x3, x4, . . . ) (xn+2) (x3, x4, x5, . . . ). , (xn+m) (x1+m, x2+m, x3+m, . . . ). 2.1,
limn+ xn = x limn+ xn+m = x.
: 1n+3 0 1n 0.
6 .7 . > 0
0 > 0.
31
-
. 2.2 :
.
2.2. xn x R u, l.[] x > u, xn > u.[] x < l, xn < l.[] u < x < l, u < xn < l.
. [] > 0 Nx() u. xn Nx(), xn > u.[] > 0 Nx() l. xn Nx(), xn < l.[] > 0 Nx() u, l. xn Nx(), u < xn < l. . [] [] xn > u xn < l. xn > u xn < l , , u < xn < l.
2.3 . .
2.3. .
. (xn) a, b. > 0 Na() Nb() . xn Na() , , xn Nb(). xn Na() xn Nb() . . (xn) a, b c a, b. [] [] 2.2 xn < c xn > c. xn < c xn > c .
2.4. [] xn x R, (xn) .[] xn +, (xn) .[] xn , (xn) .
. [] x, (x1, x+1). (xn) n0 (x1, x+1). , (xn) (x1, x+1) x1, . . . , xn01. , , , (x 1, x + 1) [l, u] (xn). (xn) .[] +, (1,+]. (xn) n0 (1,+]. (xn) (1,+] x1, . . . , xn01. , , , (1,+] [l,+] (xn). (xn) ., l xn > l. l (xn), (xn) .[] [].
32
-
2.3.3. H ((1)n1) . - 2.4[].
2.3.4. ( (1+(1)n1)n
2
), (1, 0, 3, 0, 5, 0, 7, . . . ),
. , +, , 2.2, > 1, ., (1, 0,3, 0,5, 0,7, . . . ) , . 2.4[,] .
. 2.5 :
. 2.5 2.2. , . 2.5 - : - , .8
2.5. [] xn l n xn x R, x l.[] xn u n xn x R, x u.[] u < l xn u n xn l n, (xn) .
. [] x < l, , 2.2, xn < l, xn l n . x l.[] .[] (xn) , , [] [], u l , , l u . (xn) .
2.3.5. xn x R xn [l, u] n, x [l, u].
2.3.6. ((1)n1n), (1,2, 3,4, 5,6, . . . ), , 1 1.
2.3.7. (n 3[n3 ]
), (1, 2, 0, 1, 2, 0, 1, 2, 0, . . . ), ,
2 0.
2.6. xn yn n xn x R yn y R, x y.
. ( ) y < x. a y < a < x. yn < a a < xn. , yn < a a < xn. yn < xn , , xn yn n. . x y.
2.3.8. 1n 0. xn > M , yn xn, xn > M yn xn. yn > M . ,yn +.[] .
2.3.9. 2n + (1)nn = 2n n n n n +. ,2n+ (1)nn +.
2.3.10. n2+2n+1n+2 n n n +. n2+2n+1
n+2 +.
2.3.11. n! n n. n! +.
2.8 .
2.8. xn yn zn. xn a zn a, yn a.
. > 0. xn (a , a + ) , , zn (a , a + ). xn, zn (a , a + ) , yn xn, zn, yn (a , a+ ). yn a.
2.3.12. 1n (1)n
n 1n n.
1n 0
1n 0, -
(1)n
n 0.
2.3.13. 0 n2[n/2]n 1n n. 0 0
1n 0,
n2[n/2]n 0.
.
.
. (xn) (xn). (xn) (yn) (xn + yn). (xn) (yn) (xn yn). xn yn =xn + (yn), (xn) (yn) (xn) (yn). (xn) (yn) (xnyn). (xn) (xn). , (xn) () (xn) (xn)
(1xn
).
(
1xn
) xn 6= 0 n.
(xn) (yn) (xnyn
). xnyn = xn
1yn,
(xn) (yn) (xn) (yn).
(xnyn
) yn 6= 0 n.
(xn) (|xn|).
34
-
.
2.9. x, y R .[] xn x, xn x.[] xn x, |xn| |x|.[] xn x yn y x+ y , xn + yn x+ y.[] xn x yn y x y , xn yn x y.[] xn x yn y xy , xnyn xy.[] xn x x , xn x.[] xn 6= 0 n. xn x 1x (, x 6= 0), 1xn
1x .
[] yn 6= 0 n. xn x yn y xy , xnyn
xy .
. [] xn x x R. > 0 |xn x| < ,,
|(xn) (x)| = |xn x| < .
xn x. xn +. M > 0 xn > M , , xn < M . xn , xn (+). xn . M > 0 xn < M , , xn > M . xn +, xn ().[] xn x x R. > 0 |xn x| < , ,|xn| |x| |xn x| < . |xn| |x|. xn +. M > 0 xn > M , , |xn| > M . |xn| +, |xn| |+|. xn . M > 0 xn < M , , |xn| > M . |xn| +, |xn| | |.[] xn x yn y x, y R. > 0 |xn x| < 2, , |yn y| < 2 , ,
|(xn + yn) (x+ y)| = |(xn x) + (yn y)| |xn x|+ |yn y| < 2 +2 = .
xn + yn x+ y. xn + yn y y (,+]. (yn) , l yn > l n. , M > 0 xn > M l, ,
xn + yn > (M l) + l =M.
xn + yn +, xn + yn x+ y. [].[] [] [].[] xn x yn y x, y R. (yn) , M 0 |yn| M n. , > 0 |xn x| < 2M+1 ,, |yn y| < 2|x|+1 , ,
|xnyn xy| = |(xn x)yn + x(yn y)| |xn x||yn|+ |x||yn y| M2M+1 +|x|
2|x|+1 < .
35
-
xnyn xy. xn + yn y y (0,+]. l 0 < l < y. l < yn. , M > 0 xn > Ml , ,
xnyn >Ml l =M.
xnyn +, xnyn xy. [].[] [].[] xn x x R\{0} xn 6= 0 n. |xn| |x|. |x| > 0, l 0 < l < |x| |xn| > l. , > 0 |xn x| < |x|l , , 1
xn 1x
= |xnx||x||xn| < |x|l|x|l = . 1xn
1x .
xn +. > 0 xn > 1 , , 1xn
0 = 1xn < .
1xn 0, 1xn
1+ . xn [].[] [] [].
2.3.14. xn x, x R k N, xnk xk., xnk = xn xn (k ) x x (k ) = xk. , n1n 1 (
n1n )
3 13 = 1. xn +, , , xnk +. , xn , xnk +, k , xnk , k ., ( ), , xn , 1xnk 0.:
nk
{+, k Z, k > 00, k Z, k < 0
2.3.15.(1)n1 = 1 1 ((1)n1) . ,
2.9[] ., , x = 0, : xn 0 |xn| |0| = 0., |xn 0| <
|xn| 0 < xn 0 |xn| 0 .
2.3.16. a0 + a1x+ + akxk . -, k 1 ak 6= 0.
a0 + a1n+ + aknk = nk(a0
1nk
+ a11
nk1+ + ak1 1n + ak
).
ak, 0. ,nk +.
a0 + a1n+ + aknk ak(+) =
{+, ak > 0, ak < 0
36
-
., limn+(a0 + a1n+ + aknk) = limn+ aknk. : 3n2 5n+ 2 + 12n
5 + 4n4 n3 .: 2n5 2n2 + n 7 , (2n5 2n2 + n 7)8 +.: n3 + 2n 1 , (n3 + 2n 1)5 .
2.3.17. a0+a1x++akxk
b0+b1x++bmxm , ak 6= 0, bm 6= 0.
a0+a1n++aknkb0+b1n++bmnm =
nk
nm
(a0
1nk
+ + ak1 1n + ak)/(
b01nm + + bm1
1n + bm
).
ak bm. n
k
nm = nkm,
a0+a1n++aknkb0+b1n++bmnm
(ak/bm)(+), k > mak/bm, k = m0, k < m
. : n
32n2+n+12n23n1 +,
n2+nn+2 ,
n4n3n4+1
1 n2+n+4n3+n2+5n+6
0.: 2n
3+n2+n+12n+3 ,
(2n3+n2+n+12n+3
)7 .: n
3+n+73n3+n2+1
13 ,
(n3+n+73n3+n2+1
)3 127 . 2.3.18.
na
{+, a > 00, a < 0
a > 0. M > 0 n0 na > M n n0., na > M n > M1/a. 1.1, n0 > M
1/a. , n > M1/a n0 n n0. , n0, n > M1/a , , na > M n n0. ( ) n0 = [M1/a] + 1 , n0 n0, n > M1/a n n0. a < 0. , , na 0 , : a > 0 , ,na = 1
na 1
+ = 0.
2.3.19. (a, a2, a3, a4, . . . ), (an). : a. a = 1, (1) 1. , a = 0, (0) 0. a 1, (an) a 1, a2 1, a3 1, a4 1, . . . . , 1 1 , , . a > 1. Bernoulli 1.1
an n(a 1) + 1
n. n(a 1) + 1 +, an +., , . M > 0 n0
37
-
an > M n n0. , an > M n > logaM . n0 > logaM , n0, n > logaM, , an > M n n0., 0 < |a| < 1. 1|a| > 1, |an| = 1(1/|a|)n
1+ = 0. a
n 0.:9
an
+, a > 1 1, a = 1 0, 1 < a < 1 , a 1
2.3.20. (1 + a + a2 + + an1 + an) a. :
1 + a+ a2 + + an
+, a 1 1/(1 a), 1 < a < 1 , a 1
a 1, 1 + a+ a2 + + an 1 + 1 + 1 + + 1 = n+ 1.
n+ 1 +, 1 + a+ a2 + + an +. . 1 < a < 1,
1 + a+ a2 + + an = an+11a1 01a1 =
11a .
a 1. an+1 = 1 + (a 1)(1 + a + a2 + + an) n. 1+ a+ a2 + + an x R, an+1 1+ (a 1)x. , (an+1) . (1 + a+ a2 + + an1 + an) .
2.3.21.
loga n
{+, a > 1, 0 < a < 1
a > 1. M > 0 n0 loga n > M n n0. a > 1, loga n > M n > aM . , n0 n0 > aM , , n0, loga n > M , ,n > aM n n0. 0 < a < 1. 1a > 1, loga n = log1/a n (+) = .
- - .
2.3.22.
na 1 a > 0.
a = 1 : n1 = 1 1.
a > 1. Bernoulli (1+ a1n )n 1+na1n = a
, , 1 na 1 + a1n n. , n
a 1.
0 < a < 1, 1a > 1, na = 1
n
1/a 11 = 1.
9 2.4.9 2.3.22 2.3.25.
38
-
2.3.23.
nn 1.
Bernoulli (1 +n1n )
n 1 + nn1n =
n ,
, 1 nn (1 +
n1n )
2 < (1 + 1n)2 n. n
n 1.
2.10 .
2.10. 10 xn > 0 n.[] 0 < b < 1 xn+1xn b, xn 0.[] b > 1 xn+1xn b, xn +.[] 0 a < 1 xn+1xn a, xn 0.[] a > 1 xn+1xn a, xn +.
. , .[] , n0 xn+1xn b n n0. n n0 + 1
0 < xn =xn
xn1xn1xn2
xn0+2xn0+1xn0+1xn0
xn0 b b b b xn0 = bnn0 xn0 =xn0bn0 b
n = c bn,
c = xn0bn0 . 0 < b < 1, bn 0. xn 0.
[] , n0 xn+1xn b n n0. n n0 + 1
xn =xn
xn1xn1xn2
xn0+2xn0+1xn0+1xn0
xn0 b b b b xn0 = bnn0 xn0 =xn0bn0 b
n = c bn,
c = xn0bn0 . b > 1, bn +. xn +.
[] b a < b < 1. xn+1xn b [] xn 0.[] b a > b > 1. xn+1xn b [] xn +.
2.3.24.
an
nk + a > 1 k N.
an+1/(n+1)k
an/nk= a
(n
n+1
)k a a > 1. annk
+. : (an) a > 1 (nk).
2.3.25.
an
n! 0.
a = 0, , , an
n! = 0 0. a 6= 0. |a|
n+1/(n+1)!|a|n/n! =
|a|n+1 0.
|a|nn! 0 , ,
an
n! 0. : (an) a > 1 (n!).
10 2.4.11.
39
-
2.11 2.9 .
2.11. [] xn + (yn) , xn + yn +. xn (yn) , xn + yn .[] xn 0 (yn) , xnyn 0.[] xn + (yn) , xnyn +. xn (yn) , xnyn .[] xn 6= 0 n. xn 0 (xn) , 1xn +. xn 0 (xn) , 1xn .[] xn 6= 0 n. |xn| +, 1xn 0.
. [] xn + (yn) , l yn l n. M > 0 xn > Ml , ,xn + yn > (M l) + l =M . xn + yn +. .[] xn 0 (yn) , M 0 |yn| M n. > 0 |xn| < M+1 , ,
|xnyn| = |xn||yn| MM+1 < .
xnyn 0.[] xn + (yn) , l > 0 yn l. M > 0 xn > Ml , , xnyn > Ml l =M . xnyn +. .[] xn 0 xn > 0. M > 0 |xn 0| < 1M . 0 < xn M . 1xn +.
.[] xn 6= 0 n |xn| +. 1|xn| 0 , ,
1xn
0.
2.3.26. (xn) , xnn 0. , 1+(1)
n1
n 0. :n3[n/3]
n 0 sinnn 0.
2.3.27. ((1)n1n
)
(1)n1n = n +. 1
(1)n1n 0. ,1
(1)n1n =(1)n1
n 0. ((1)n1n
) .
x = 0 2.9[]; 10 .
2.3.28. (1)n1
n 0, ((1)n1n
)
.
2.3.28;
( (1)n1n
), ,
. (xn) , ( 1xn ) +. , (xn) , ( 1xn ) . , (xn) , . xn 0
40
-
|xn| 0 1|xn| +. 1|xn| 1, ,
, 1xn 1 n 1xn
1 n. ( 1xn ) .
, , : 10 0 0 .
0+ 0 0 0 , , 2.11[].
. 10+ = +,
10 = .
1.1 : , , , .
, 2.12, 4.3. , ( ) . - 1.4 ab. 00, 1+,1, (+)0 0.
. (xn) (yn) (xn
yn).
2.12. xn > 0 n. xn x R yn y R xy , xnyn xy. , xn 0 yn , xn
yn +.
2.3.29. 2.12, na 1 a > 0.
, (a) ( 1n). a a 1n 0,
na = a1/n a0 = 1.
2.3.30. (an) a > 1 0 < a < 1 ,, 2.12. (a) (n). a a n +, an a+, +, a > 1, 0, 0 < a < 1.
2.3.31. nn 1 2.12.
(n) ( 1n), n + 1n 0, (+)
0 .
0 , , , - 2.12 .
. (0+) = +.
.
2.3.1. ( (n+1)27(n+3)79
(2n+1)106
),(n3+(1)nn2+1
3n2+2(1)n1n),(n(n+1)
n+4 4n3
4n2+1
),
((1 n)5 + n4),(( n
3+n+13n2+3n+1
)9),( 3n+(2)n3n+1+2n+1
), (n+ 1
n), (
n2 + n+ 1
n2 + 1).
41
-
2.3.2. , , (1+ 2+22 + +2n),(1+ 12 + +
12n
),
(1 2 + 22 + + (1)n2n),(27
37+ 2
8
38+ + 2n+6
3n+6
),(2n
3n +2n+1
3n+1+ + 22n
32n
).
2.3.3. (212+1
313+1
n1n+1
),(23123+1
33133+1
n31n3+1
).
2.3.4. x 6= 1 (xn1xn+1) .
x (x2n1
x2n+1) .
2.3.5. x limn+ (x+1)2n
(2x+1)n ;
2.3.6. x 6= 1 xn 6= 1 n. xn x xn
1xn x
1x .
2.3.7. 11 3+(1)n
2n 0, 3+(1)n
2n > 0 n (3+(1)n2n
) .
(3(1)n1)n
2 + ( (3(1)n1)n
2
) .
2.3.8. (xn) n2 2n < n2xn n2 + 3. : n + 1 2nxn n + 2xn + 3, n2 + nxn 15n n2xn
2 2n(n 1)xn + n2 2n 3 0.
2.3.9. xn x R yn y R. x < y, xn < yn.
2.3.10. xn x R yn y R x 6= y. |x y| > a, |xn yn| > a.
2.3.11. , xn [l, u] n xn x, x [l, u]. x (xn), xn (l, u) n; ;
2.3.12. 2.5[], -
(2(1)
n1), ((1 + (1)n12 )n), ((1)n1 + 10n3 ), ((1)n1 nn+1).2.3.13. 2n+ (1)n1n +, 22n+(1)n1n 0,
(12 +
(1)n14
)n 0.2.3.14.
[3n2n+1
n+2
] +, [
n]n
1 n+23n2n+1
[3n2n+1
n+2
] 1.
2.3.15. (1 + 1n)n2 +, (1 1
n2)n 1, (1 + 1
n2)n 1.
2.3.16. 12 [nx]
+, x > 00, x = 0, x < 0
[nx] [ny]
+, x > y0, x = y, x < y
nx [ny]
+, x > y 0, x = y Z , x < y , x = y Q \ Z
2.3.17. (2n)!(n!)2
+.
2.3.18. 0 a b c. nan + bn b, n
an + bn + cn c.
11 0 +, - , , .
12 x = y R \Q 2.7.18.
42
-
2.3.19. nn3 1, n
n4 + 3n2 + n+ 1 1.
2.3.20. a, [a]+[2a]++[na]n2
a2 .
2.3.21. nn2+1
+ nn2+2
+ + nn2+n
1 1n2+1
+ 1n2+2
+ + 1n2+n
1.
2.3.22. limm+(limn+(cosm!x)2n
)=
{1, x Q0, x R \Q
2.3.23. (xn) - : xn+1 = xn+2, xn+3 = xn3, xn+1 = xn22, xn+2 = xn2+2,xn+1 = xn
2 + 3, xn+2 = xn+1 + xn3 ;
2.3.24. xn x yn y, max{xn, yn} max{x, y} min{xn, yn} min{x, y}.
2.3.25. : n 1n =1n + +
1n (n ) 0 + + 0 (n ) = n0 = 0.
: (1 + 1n)n = (1 + 1n) (1 +
1n) (n ) 1 1 (n ) = 1
n = 1. 2.9[,];
2.3.26. x1 > 0 xn+1 x1 + + xn n N. 0 < a < 2, xnan +. a = 2 (2
n).
2.3.27. (xn), (yn) (xn + yn) . (xn), (yn) (xnyn) .
2.3.28. (xn + yn) (xn), (yn) , , , . (xnyn) (xn), (yn) , , , .
2.3.29. 13 (xn), (yn) xn +, yn (xn + yn) (i) c R (ii) . (xn), (yn) xn 0, yn + (xnyn) (i) c R (ii) . (xn), (yn) xn 0, yn 0 (xnyn ) (i) c R (ii) . (xn), (yn) xn +, yn + (xnyn ) (i) c [0,+] (ii) . (xnyn ) c [, 0);
2.3.30. 14 (xn), (yn) xn +, yn 0 (xn
yn)(i)
c [0,+] (ii) . (xn
yn)
c [, 0); (xn), (yn) xn 0, yn 0
(xn
yn)(i) -
c [0,+] (ii) . (xn
yn) c [, 0);
(xn), (yn) xn 0, yn (xn
yn)(i)
c {+,} (ii) . (xn
yn) c R;
2.3.31. (xn), (yn) xn, yn > 0 n, xn 0, yn + (xnyn) .
13 .14 . 2.4.10.
43
-
2.3.32. x (rn) rn x. x (tn) tn x. x (rn) (sn) rn x sn x. .
2.3.33. - A. A supA A supA. infA. [0, 2], [0, 2), {2}, [0, 1]{2} supremum . , ( ) , , . , , , . N, Z, Q, { 1n |n N}, (0, 1] (R \ Q), (0, 2) Q , supremum infimum .
2.3.34. - A u A. u = supA A u. infA l A.
2.3.35. - A. supA A A supA. supA / A, A supA. infA.
2.3.36. k N, k 2 xn 0 n. xn x, k
xn k
x.
xn +, kxn +.
2.12.
2.3.37. |xnxm| 1 n,m n 6= m. |xn| +. (xn) ; (n), (n), ((1)n1n).
2.3.38. xn x xn x n, sup{xn |n N} = x. x < y xn < y n xn x, sup{xn|n N} < y.
2.3.39. xn x, k inf{xn |n k} x sup{xn |n k}.
2.3.40. xn x. (xn) k xk x. (xn) k xk x.
2.3.41. 15 [] Cesro: xn x R, x1++xnn x. xn = (1)n1 n, x1++xnn 0. xn =
1+(1)n2 n n,
x1++xnn +.
(xn) . - Cesro.[] an+1 an a R, ann a.
15 2.7.13.
44
-
[] Cesro: 16 (xn), (yn) yn > 0 n y1 + + yn +. xnyn l R,
x1++xny1++yn l.
[] xn > 0 n xn x [0,+], nx1 xn x.
[] an > 0 n an+1an a [0,+], nan a.
( nn),
(nn!)
(n
(2n)!/(n!)2).
2.4 .
2.1 . , , , , . - : ((1)n1) ((1)n1n) .
2.1 .
2.1. 17 . :[] (xn) , limn+ xn = sup{xn |n N}.: (xn) , +, , . - .[] (xn) , limn+ xn = inf{xn |n N}.: (xn) , , , . - .
. [] - {xn |n N}. ( ) supremum , ( ) supremum +. (xn) .
sup{xn |n N} = +
xn +.M > 0. M {xn |n N}, n0 xn0 > M . (xn) ,
xn xn0 > M
n n0. xn +. (xn) .
x = sup{xn |n N}
xn x. > 0. x < x, x {xn |n N}. n0 x < xn0 . (xn) ,
x < xn0 xn16 Cesro yn = 1 n.17 supremum 2.4.17.
45
-
n n0. , x {xn |n N},
xn x < x+
n. x < xn < x+
n n0. xn x.[] .
2.1. (xn) - , , 2.1, (xn) , x, . , xn x n. , , (xn) , xn < x n., xn0 = x n0, ( ) x = xn0 xn x n n0 , , . -, (xn) , xn < x n. . : (xn) xn x, xn x n. , , (xn) (, , ), xn < x n. (xn) xn x, xn x n. , , (xn) (, , ), xn > x n.
2.1 . , , ( ) .18
2.4.1. (xn) x1 = 1
xn+1 =3xn+6xn+4
n 1.
x1 = 1, x2 =95 , x3 =
5729 , x4 =
345173 .
, , 4, . , ,
xn 0
n. , x1 = 1 0 , xn 0, , , xn+1 0. , - xn+1 xn :
xn+1 xn = 3xn+6xn+4 3xn1+6xn1+4
= 6(xnxn1)(xn+4)(xn1+4) .
xn+1xn n , x2x1 > 0, xn+1 xn > 0 n. (xn) . xn+1 > xn 3xn+6xn+4 > xn , ,xn
2 + xn 6 < 0 , , 3 < xn < 2 n. , (xn) 18, , 2.3.23.
46
-
, , ., , xn x. xn 0 n x 0 , ,
x = 3x+6x+4 .
x = 3 x = 2 , x 0, x = 2. , xn 2., (xn) , xn < 2 n. . : : xn 0 n (xn) ( ), (xn) . xn +, , xn+1 =
3+(6/xn)1+(4/xn)
n, + = 3, . (xn) , , xn 2. : . x1 = 1, x2 = 1.8, x3 = 1.9655... x4 = 1.994219... . (xn) 2, xn < 2 n. , , , (xn) 2.
.
2.4.2. (1 + 11! +
12! + +
1n!
).
xn = 1 + 11! +12! + +
1n! n.
xn+1 = 1 +11! +
12! + +
1n! +
1(n+1)! = xn +
1(n+1)! > xn
n, (xn) .,
n! 2n1
n. , n = 1 n 2 n! =1 2 3 n 1 2 2 2 = 2n1.,
xn 1 + 11 +12 + +
12n1 = 1 +
1(1/2n)1(1/2) < 1 +
11(1/2) = 3.
n. (xn) , , .
2.4.3. ((1 + 1n)
n) , ,
.19
an =(1 + 1n
)n n. Newton ( 1.1) x = 1n y = 1,
an = 1 +(n1
)1n +
(n2
)1n2
+ +(nk
)1nk
+ +(nn
)1nn
= 1 + 11! +12!(1
1n) + +
1k!(1
1n)(1
2n) (1
k1n )
+ + 1n!(11n) (1
n1n ).
(2.2)
, , n+ 1,
an+1 = 1 +(n+11
)1
n+1 +(n+12
)1
(n+1)2+ +
(n+1k
)1
(n+1)k+ +
(n+1n
)1
(n+1)n
+(n+1n+1
)1
(n+1)n+1
= 1 + 11! +12!(1
1n+1) + +
1k!(1
1n+1)(1
2n+1) (1
k1n+1)
+ + 1n!(11
n+1) (1n1n+1) +
1(n+1)!(1
1n+1) (1
n1n+1)(1
nn+1).
(2.3)
19 2.4.4.
47
-
(2.2) (2.3). k 2 k n, k- (2.2) k- (2.3), n n+1. , (2.3) , k = n+ 1.
an < an+1
n, (an) . xn = 1 + 11! +
12! + +
1n! n, .
(xn) xn < 3 n. (2.2) > 0 < 1,
an 1 + 11! +12! + +
1k! + +
1n! = xn < 3 (2.4)
n. (an) , , ,, . a (an).
an a.
, , k (2.2) , () k-,
an 1 + 11! +12!(1
1n) + +
1k!(1
1n)(1
2n) (1
k1n ).
n +, ( k)
a 1 + 11! +12! + +
1k! = xk.
, , k , , a xn n. (2.4)
an xn a
n , an a,
xn a.
. 2.4.2 2.4.3, (1 + 11! +
12! + +
1n!
)
((1 + 1n)
n) .
e. ,
e = limn+(1 + 1n
)n= limn+
(1 + 11! +
12! + +
1n!
).
e
(1 + 11! +
12! + +
1n!
)
((1 + 1n)
n). e
, e . ((1 + 1n)
n) , (1 + 1n)n < e n.
, 1 + 11! +12! + +
1n! < e n.
, 1+ ( 1.4 2.3). xn 1 yn + :xnyn 1. : xnyn 1yn = 1 1 . , , (1+ 1n)
n e . , 1 + 1n 1 n + (1 +
1n)
n 1.
. e - y > 0, loge y,
log y ln y.
48
-
2.13 , , 1.9.
2.13. [] log(y1y2) = log y1 + log y2 y1, y2 > 0.[] log(yz) = z log y y > 0 z.[] loga y =
log ylog a y > 0 a > 0, a 6= 1.
[] log 1 = 0, log e = 1.[] 0 < y1 < y2, log y1 < log y2.
.
2.4.4. 20
1 + 12 +13 + +
1n +.
xn = 1 + 12 +13 + +
1n n.
xn+1 xn = 1n+1 > 0 n, (xn) , , . n
x2n xn = 1n+1 + +1
n+n 1
n+n + +1
n+n = n1
n+n =12 .
(xn) x, , (xn) , xn x2n x x. xn x, x2n x. x2n xn x x = 0 , x2n xn 12 n. xn +. xn + . , n k 0 2k n < 2k+1., n 2. , - k log2 n < k + 1, k = [log2 n]. , k,
xn = 1 +12 + (
13 +
14) + (
15 +
16 +
17 +
18) + + (
12k1+1
+ + 12k) + 1
2k+1+ + 1n
1 + 12 + (14 +
14) + (
18 +
18 +
18 +
18) + + (
12k
+ + 12k)
= 1 + 12 + 214 + 4
18 + + 2
k1 12k
= 1 + 12 +12 +
12 + +
12 = 1 +
k2 .
xn 1 + 12 [log2 n] > 1 +
12(log2 n 1) =
12 log2 n+
12
n , log2 n +, xn +.
2.4.5. 21
(1 + 1
22+ 1
32+ + 1
n2
).
xn = 1 + 122 +132
+ + 1n2
n. xn+1xn = 1(n+1)2 > 0 n, (xn) .
xn 1 + 112 +123 + +
1(n1)n = 1 + (
11
12) + (
12
13) + + (
1n1
1n) = 2
1n < 2
n. (xn) , , .20 2.5.4, 2.6.2 7.3.20
8.2.7, 8.2.10 8.3.1. 2.4.6, 6.4.11.21 2.6.1, 8.2.7 8.2.10 6.4.11, 7.3.20 8.2.1.
49
-
. [a1, b1], [a2, b2], . . . [an+1, bn+1] [an, bn] n. , (an) - (bn) an bn n. :(i) (an) (bn) .(ii) x an x bn n.(iii) x (ii) bn an 0. (iii), (an), (bn) x .
. (an) (bn) ,
a1 an bn b1
n, (an) , , ( b1 ) (bn) , , ( a1 ).
an a, bn b.
an bn n, a b. ,
an a b bn n. x [a, b] an a x b bn n. , x an x bn n, a x b, x [a, b]. x an x bn n - [a, b]. , x [a, b] , , a = b , , bn an 0. x x = a = b.
an x bn , , x [an, bn]. , an x bn n x [an, bn], . : [an, bn] , +
n=1[an, bn] = [a, b],
a = limn+ an b = limn+ bn.
2.4.6. p N, p 2. x [0, 1),
0 x < 1. (2.5)
, x p [0, 1p
),[1p ,
2p
), . . . ,
[p1p , 1
) 1p . x; kp x 1, (xn) . x1 < 1 x1 6= k1k k N, (xn) . x1 = k1k k N;[] x1, x2 > 0 xn+2 = xn+1 + 2xn n. (xn+1xn ) .[] xn+1 = sinxn n. (xn) , , .
2.4.9. 28 [] a > 1, (an) , an+1 = aan, an +. 0 < a < 1.[] a > 1 k N, (an
nk) , -
an+1
(n+1)k= a n
k
(n+1)kan
nk, a
n
nk +.
[] a > 1, ( na) , -
(
2na)2
= na, n
a 1. a = 1, 0 < a < 1;
[] ( nn)
nn 1.
[] (an
n! ) , an+1
(n+1)! =an
n!a
n+1 , an
n! 0.
2.4.10. 29 (xn), (yn) xn 1, yn + (xn
yn)(i)
c [0,+] (ii) . (xn
yn)
c [, 0); (xn), (yn) xn 1, yn
(xn
yn)(i)
c [0,+] (ii) . (xn
yn) c [, 0);
2.4.11. [] b , 1b+1+1
b+2+1
b+3+ +1
b+n +.[]30 (1 + a1) (1 + an) 1 + a1 + + an, a1, . . . , an 0, (1 a1) (1 an) 1 a1 an, 0 a1, . . . , an 1. b , limn+
(a+1)(a+2)(a+n)(b+1)(b+2)(b+n) , -
a = b, a > b, a < b.[] xn > 0 n. c < 0 n
(xn+1xn
1) c, xn 0.
c > 0 n(xn+1
xn 1
) c, xn +.
27 2.5.5.28 2.3.19 2.3.22 2.3.25.29 . 2.3.30.30 (1 + a)n 1 + na Bernoulli a 0
1 a 0, .
53
-
c < 0 n(xn+1
xn 1
) c, xn 0.
c > 0 n(xn+1
xn 1
) c, xn +.
2.10. (2n)!
4n(n!)2 0 enn!nn +.
2.10;
2.4.12. 31 y 0 k N, k 2. (xn) x1 > 0 xn+1 = k1k xn +
1k
yxnk1
n. xn > 0 n., Bernoulli, xnk y n 2, , xn+1 xn n 2. (xn) , x = limn+ xn, xk = y x 0.
2.4.13. xn+1 xn+xn+22 n.32
, , (xn) . (xn xn+1) xn xn+1 0. (xn) . (xn) . (xn) , , , .
2.4.14. 0 < a b. x1 = a y1 = b xn+1 =
xnyn yn+1 = xn+yn2 n,
(xn) , (yn) , xn yn n (xn), (yn) , GA (a, b).33
w1 = a z1 = b wn+1 = 2wnznwn+zn zn+1 =wnzn n,
(wn) , (zn) , wn zn n (wn), (zn) , HG (a, b).34
a, b,HG (a, b),GA (a, b) H (a, b) = 2aba+b , G (a, b) =
ab A (a, b) = a+b2 ;
2.4.15. xn = 135(2n1)246(2n) n. (nxn2)
((n+ 12)xn
2) .
.
2.4.16. [] f : [a, b] R [a, b]. f(a) > a f(b) < b, (a, b) f() = .[] I f : I R x I > 0 f(x) f(x) f(x) x, x (x , x + ) I x < x < x. f I .
2.4.17. . - , supremum.35
. 12n 0., - A. x1 A y1 A. [x1, y1] A A. x1+y12 A, x2 = x1, y2 =
x1+y12 , ,
x2 =x1+y1
2 , y2 = y1. [x2, y2] A 31 1.2.32 (xn) . n, -
.33 GA (a, b) - a, b.34 HG (a, b) - a, b.35 supremum : .
54
-
A. , [x1, y1], [x2, y2], . . . [xn+1, yn+1] [xn, yn] yn xn = y1x12n1 n [xn, yn] an A un A. u xn u, yn u , , an u, un u. u A.
2.4.18. 36 I , . ( ) (an) I = {an |n N}. [x1, y1] I y1 x1 > 0 a1 / [x1, y1]. [x2, y2] [x1, y1] y2 x2 > 0 a2 / [x2, y2]. , [x1, y1], [x2, y2], . . . [xn+1, yn+1] [xn, yn] an /[xn, yn] n. [xn, yn] , , 6= an n. I .
2.4.19. 37 1 , n 2, 2n 2n . pn qn , , . p2 = 4
2 q2 = 8
pn+1 = 2pn(2 +
(4 pn
2
4n
)1/2)1/2 qn+1 = 4qn(2 + (4 + qn24n )1/2)1 n 2. qn = pn
(1 pn
2
4n+1
)1/2 n 2. (pn) , (qn) pn < qn n 2. (pn), (qn) .38
2.4.20. 39 x (rn) rn x. , p- x - 2.3.32. . y > 1 x . (rn) rn x (yrn) . - (rn) rn x (yrn) . yx = limn+ yrn (rn) rn x. yx = limn+ yrn ( ) (rn) rn x. y > 1 x . yx = limn+ yrn ( ) (rn) rn x. 0 y 1 x . yx - ( 1.4). 1.8 .
2.4.21. 40 y > 0.
(n( n
y 1)
) .
. log y = limn+ n( n
y 1).
36 A, R, a : N A A. an = a(n), A A = {an |n N} , , A . A , .
37 .38
, , , 2, . , pn 2 qn n 2. pn 2 qn 2.
39 .40 .
55
-
log y = limn+ yxn1xn
(xn) xn 0 xn 6= 0 n. 2.13 ( []) . a > 0, a 6= 1 y > 0. loga y =
log ylog a 1.9.
2.5 .
. (xn). n1, n2, n3, . . . , nk, . . . n n1 < n2 < < nk < nk+1 < . (xn). x1, x2, . . . , xn, . . . xn1 , xn2 , . . . , xnk , . . . . : xn1 , xn2 . , (xnk). , (xnk) (xn).
, n1 < n2 < < nk < nk+1 < , .
2.5.1. n1 = 2, n2 = 5, n3 = 6, n4 = 9, n5 = 13, (xn) x2, x5, x6, x9, x13., n1 = 2, n2 = 5, n3 = 6, n4 = 10, n5 = 8 (xn). x2, x5, x6, x10, x8 (xn) : x10 x8 (xn) ( x9) x10 x8 .
.
2.5.2. nk = 2k k, (xn), (x2k) (x2, x4, x6, x8, x10, . . . ).
2.5.3. nk = 2k1 k, (xn), (x2k1) (x1, x3, x5, x7, x9, . . . ).
2.5.4. nk = k k, (xk) (x1, x2, x3, x4, x5, . . . ). (xn) (xn).
2.5.5. nk = 2k1 k, (x2k1) (x1, x2, x4, x8, x16, . . . ).
(xnk) k. k - 1, 2, 3, . . . , nk (xn).
, (xn) (xn). , , (xn) (xn). , (xn), (xn) . - (xn) n1. , xn1 , (xn) n2. , xn1 xn2 , (xn) n3 . (xnk) (xn) (xn) .
2.2. nk N nk < nk+1 k. nk k k.
56
-
. n1 1 n1 N. nk k k. nk+1 > nk nk, nk+1 N, nk+1 nk + 1 , , nk+1 k + 1. nk k k.
2.2
nk +.
2.14. , .
. xn x R (xnk) (xn). xnk x. > 0. n0 xn Nx() n n0. nk +. k nk n0 , , xnk Nx(). xnk x.
2.14 , , : , .41
2.5.6. ((1)n1) ., (1)(2k1)1 = 1 1 (1)(2k)1 = 1 1.
2.15 .
2.15. x R x2k x x2k1 x. xn x.
. > 0. x2k x, xn Nx() n n . , x2k1 x, xn Nx() n n . xn Nx() n ( n n) . xn x.
2.5.7. 42
(1 12 +
13
14 + + (1)
n1 1n
).
xn = 1 12 +13
14 + + (1)
n1 1n n.
x2k+2 x2k = 12k+1 1
2k+2 > 0 k. ,
x2k = 1 (12 13) (
14
15) (
12k2
12k1)
12k < 1
k, . (x2k) , , ., x2k+1 x2k1 = 12k +
12k+1 < 0 k. ,
x2k1 = (1 12) + (13
14) + + (
12k3
12k2) +
12k1 > 0
k, . (x2k1) - , , .,
x2k x2k1 = 12k 0,
(x2k), (x2k1) . (xn) . x (xn), :
x2 < x4 < . . . < x2n < x2n+2 < . . . < x < . . . < x2n+1 < x2n1 < . . . < x3 < x1.
41 . 2.5.15.42 2.6.3 6.4.11 8.3.9.
57
-
(x2n) x , , (x2n1) x. , , [x2, x1],[x4, x3], [x6, x5], . . . x - .
. . , ((1)n1) . , ((1)n1), , : 1 1. ((1)n1) . , .
BOLZANO - WEIERSTRASS. .
. 43 (xn) l, u l xn u n. (xn) , : , , . [l, u] [l, l+u2 ], [
l+u2 , u]. ()
(xn) [l, u], (xn). [l1, u1]. [l1, u1] [l, u],u1 l1 = ul2 [l1, u1] (xn). (xn) ( ) [l1, u1]: xn1 [l1, u1]. [l1, u1] [l1, l1+u12 ], [
l1+u12 , u1]. [l1, u1] -
(xn), (xn). [l2, u2] ( ). [l2, u2] [l1, u1], u2 l2 = u1l12 [l2, u2] (xn). (xn) ( ) [l2, u2]: xn2 [l2, u2]., , n2 > n1. , (xn) [l2, u2]. [l2, u2] [l2, l2+u22 ], [
l2+u22 , u2]. [l2, u2] -
(xn), (xn). [l3, u3]. [l3, u3] [l2, u2],u3 l3 = u2l22 [l3, u3] (xn). (xn) ( ) [l3, u3]: xn3 [l3, u3]. ,, n3 > n2. . , , [lk, uk] k
[lk+1, uk+1] [lk, uk], uk+1 lk+1 = uklk2
k. , xnk (xn) k
nk+1 > nk, xnk [lk, uk]
k. uk+1 lk+1 = uklk2 k uk lk =
ul2k
k,
uk lk 0.43 2.5.9.
58
-
, (lk), (uk) . lk x uk x. nk+1 > nk k, (xnk) - (xn) , lk xnk uk k, xnk x.
+ . - . (1, 0, 3, 0, 5, 0, 7, . . . ) +., , +, +: , (1, 3, 5, 7, . . . ). .
2.16. [] - +.[] .
. 44 [] (xn) . u (xn) > u. 45
( ) u (xn) > u. (xn) n0 (, u]. (xn) (, u] x1, . . . , xn01. , , , (, u] (, u] (xn). (xn) . . (xn) +, . (xn) > 1. : xn1 > 1. (xn) > 2. : xn2 > 2., , n2 > n1. > 2. (xn) > 3. : xn3 > 3 n3 > n2. . xnk (xn) nk+1 > nk xnk > k k. (xnk) (xn) xnk +.[] .
.
2.5.1. a < b < c < d. ( ) a, b, c d. .
2.5.2. (xn) : () ,() , , , . - (xnk) (xn) .
2.5.3. 2.3.36 2.4.3 (1 + pqn)
n qep p Z q N q 2. , (1 + rn)
n er r Q. x. > 0. ex, r, s Q r < x < s ex < er < es < ex + . , ex < (1 + rn)
n < (1 + xn)n < (1 + sn)
n < ex + (1 + xn)n ex.
44 2.5.9.45 2.4[]
2.4[].
59
-
2.5.4. (xn) (xnk) xnk x R, xn x. (xn) (xnk) xnk x R, xn x.46 xn = 1 + 12 +
13 + +
1n n. 2.4.4
x2n xn 12 n. x2k k2 + 1 k ,
, xn +.
2.5.5. 2.4.7, , t 0, (1 + t1! +
t2
2! + +tn
n!
)
((1 + tn)
n) .
2.5.6. [] x1 > 0 xn+1 = 1+ 2xn n. (x2k), (x2k1) . (xn) .[] x1 > 0 xn+1 = 1 + 31+xn n. (x2k), (x2k1) . (xn) .[] 0 < p < 1 xn+2 = (1 p)xn+1 + pxn n. (x2k), (x2k1) . (xn) . xn n, yn = xn+1 xn n. (xn).
2.5.7. a, b, x R a 6= b. x2k a x2k1 b - (xnk) xnk x. (xnk) (x2k) (x2k1). (xnk) (x2k) (x2k1); x = a x = b.
2.5.8. [] x R x3k x x3k1 x x3k2 x. - 2.15, xn x. . a, b, c, x R, a 6= b, a 6= c, b 6= c. x3k a x3k1 b x3k2 c (xnk) xnk x. x = a x = b x = c.[] N - (xn). N, . - , (xn) . N . . n N n = 2m1k m N k N. N : A(m) = {2m1k | k } m N. A(1) , A(2) - , A(3) . , xn = 1k n = 2
m1k m N k N. (xn) : (x
(m)k )
+k=1 m N, ,
m N, x(m)k =1k k N. ,
0. (xn) 0, (x2m13)+m=1.
2.5.9. (xn). xn - m > n xm > xn.47
46 2.4.4. 2.6.2 7.3.20 - 8.2.7, 8.2.10 8.3.1 2.4.6 6.4.11.
47 xn xm +.
60
-
(xn) -, (xn). (xn) -, - , , . . Bolzano - Weierstrass 2.16.
2.5.10. (xn) n (xnk) k.
2.5.11. 48 , - .
2.5.12. (xn) x R (xn) x. : (xn) (xn) .
2.5.13. xn < x n. sup{xn |n N} = x (xn) x. ( 1n). sup{
1n |n N} = 1. , (
1n)
0, ( 1n), 1. - ;
2.5.14. (xn) (xnk). (xnk) (xn).
2.5.15. (xn) , . 2.14. (xn) , l, u u < l xn u n xn l n. 2.5[].
2.5.16. [] xn x xn 6= x n. (xn) .[] (rn) , rn = pnqn pn Z qn N n. qn +.[] x (rn) rn x rn = pnqn pn Z qn N n. qn + pn x(+).
2.6 .
. (xn) Cauchy > 0 n0 |xn xm| < n,m n0. :
limn,m+(xn xm) = 0.
: (xn) Cauchy .
2.17. (xn) , Cauchy.48 2.1.9.
61
-
. xn x. > 0. n0 |xn x| < 2 n n0. , n m, |xm x| < 2 m n0.
|xn xm| = |(xn x) (xm x)| |xn x|+ |xm x| < 2 +2 =
n,m n0. (xn) Cauchy.
Cauchy 2.17.
CAUCHY. (xn) Cauchy, .
. 49 (xn) Cauchy. n0 |xn xm| < 1 n,m n0. (m = n0), n n0 |xn xn0 | < 1,
|xn| = |(xn xn0) + xn0 | |xn xn0 |+ |xn0 | < 1 + |xn0 |.
, M = 1+ |xn0 |, (xn) [M,M ], (xn) . (xn) , , Bolzano - Weierstrass, - (xnk) .
xnk x.
xn x. > 0. n0
|xn xm| < 2 n,m n0.
n n0 , , . nk +, nk n0 , ,
|xn xnk | < 2 . (2.9)
, xnk x,
|xnk x| < 2 . (2.10)
(2.9) (2.10). k (2.9) (2.10) k
|xn x| = |(xn xnk) + (xnk x)| |xn xnk |+ |xnk x| < 2 +2 = .
|xn x| < n n0, xn x.
Cauchy 2.1. (xn) x (xn), |xn x| x, |xnxm| . , , Cauchy Cauchy.
49 2.6.6, 2.6.7 2.7.14.
62
-
2.6.1. (xn) xn = 1+ 122 +132
+ + 1n2
n. , ( 2.4.5.50) (xn) .m > n.
|xn xm| = 1(n+1)2 +1
(n+2)2+ + 1
(m1)2 +1m2
< 1n(n+1) +1
(n+1)(n+2) + +1
(m2)(m1) +1
(m1)m
= ( 1n 1
n+1) + (1
n+1 1
n+2) + + (1
m2 1
m1) + (1
m1 1m) =
1n
1m 0. n0 1n0 < . m > n n0 |xn xm| < 1n
1n0< .
(xn) Cauchy , , .
Cauchy R.
. Cauchy .
Bolzano - Weierstrass, - , , supremum.
.
2.6.1. (xn), (yn) Cauchy. , , (xn+yn),(xnyn) Cauchy.
2.6.2. 51 xn = 1 + 12 + +1n n, 2.4.4
x2n xn 12 n. (xn) Cauchy; xn +.
2.6.3. 52 xn = 1 12 +13
14 + +
(1)n1n n.
|xn xm| = 1n+1
1n+2 + +
(1)mn1m
1n+1 n,m n < m. (xn) Cauchy .
2.6.4. [] 0 M < 1 |xnxn+1| cMn. n0 |xn xm| c M
n
1M n,m n0 n < m. (xn) Cauchy. x (xn), |xn x| c M
n
1M .[] 0 M < 1 |xn+1 xn+2| M |xn xn+1|.53 (xn) Cauchy. x (xn), c 0 |xn x| c M
n
1M .[] x1 > 0 xn+1 = 1 + 31+xn n. (xn) . n- xn .[] || < 1 xn+1 = a+ sinxn n. (xn) . n- xn .
2.6.5. . , - , supremum.54
50 8.2.7 8.2.10 6.4.11, 7.3.20 8.2.1.51 2.4.4. 2.5.4 7.3.20
8.2.7, 8.2.10 8.3.1 2.4.6 6.4.11.52 2.5.7. 6.4.11 8.3.9.53 (xn) . 0 M 1, .54 supremum -
.
63
-
2.4.17 . , -. 12n 0 , , (xn), (yn) Cauchy. - .
2.6.6. 55 Cauchy (xn). [a1, b1] b1 a1 < 1 xn [a1, b1]. - [a2, b2] [a1, b1] b2 a2 < 12 xn [a2, b2]. , [ak, bk] k [ak+1, bk+1] [ak, bk] bk ak < 1k k xn [ak, bk] k. x x [ak, bk] k , , xn x.
2.6.7. 56 Cauchy (xn). ln = inf{xk | k n} un = sup{xk | k n} n ln un ln ln+1 un+1 un n. un ln 0 x ln x un x. ln xn un n, xn x.
2.7 .
. (xn) x R. x (xn) (xn) x.
2.7.1. (xn) , , . , ( 1n) 0 (n) .
2.7.2. (xn) xn = (1)n1 n - , 1 1. (x2k) 1 (x2k1) 1. x (xn). (xnk) (xn) x. (x2k) (x2k1) (xn), (xnk) (x2k) (x2k1). (xnk) (x2k), (xnk) (x2k) - , (xnk), x , (x2k), 1 , , x = 1., (xnk) (x2k1), x = 1., x (xn), x = 1 x = 1. (xn) 1 1.
2.7.3. , ( (1)n1+1
2 n)
0 +.
Bolzano - Weierstrass 2.16, : , - , , . - -.
55 Cauchy.56 Cauchy.
64
-
2.2. - .
. : (xn) . + (xn) , , - (xn). : (xn) . , , (xn). : (xn) 6= . (xn) , u xn u n. (xn) , , (xn) u. (xn) - 6= u. L (xn),
L = {x R |x (xn) }.
- R. L supremum -
x = supL.
x (xn). x L, L , , (xn). > 0. x = supL, x L x < x x , ,x < x < x + . x L, (xn) x. (x, x+) , , (xn) (x , x+ ). = 1, 12 ,
13 , . . . .
(xn) (x 1, x+1), , xn1 ,
x 1 < xn1 < x+ 1.
(xn) (x 12 , x+12), ,
xn2 , x 12 < xn2 < x+
12 n2 > n1.
(xn) (x 13 , x+13), ,
xn3 , x 13 < xn3 < x+
13 n3 > n2.
, (xnk) (xn)
x 1k < xnk < x+1k k
, ,xnk x.
x , , (xn). , (xn) . (xn) .
65
-
. 57 (xn) (xn)
lim supn+
xn lim supxn limxn.
(xn) (xn)
lim infn+
xn lim infxn limxn.
2.18. [] limxn :(i) limxn < x, xn < x.(ii) x < limxn, x < xn n.[] limxn :(i) x < limxn, x < xn.(ii) limxn < x, xn < x n.
. [] (i) limxn < x xn < x. (xn) x. (xn) x. , , (xn) x . x, (xn) x. , limxn < x limxn (xn).(ii) x < limxn. limxn (xn), - (xn) limxn. > x, (xn) > x.[] .
. , , , .
2.19. (xn).[] limxn limxn.[] (xn) limxn = limxn. , (xn) , limxn = limxn = limxn.
. [] , limxn limxn (xn).[] (xn) . limxn , , (xn). limxn = limxn = limxn. ( 2.7.1.), limxn = limxn. limxn = limxn = +, (i) 2.18[] x x < xn , , limxn = +. limxn = limxn = , (i) 2.18[] x xn < x , , limxn = ., limxn = limxn = x R. (i) 2.18[] > 0 xn < x+ (i) 2.18[] > 0 x < xn. > 0 x < xn < x+ . limxn = x.
57 2.7.15 .
66
-
.
2.7.1. a < b < c (a, b, a, b, a, b, a, b, . . . ) (a, b, c, a, b, c, a, b, c, . . . ). lim lim .
2.7.2. , , lim, lim : (n+1n ), ((2)n),
(2(1)
n1n),(
(1)n1(1 1n)),(2n3[n/3]
).
2.7.3. limxn = + (xn) . limxn = xn . limxn ;
2.7.4. limxn = lim(xn).
2.7.5. xn yn, limxn lim yn limxn lim yn.
2.7.6. an bn cn. lim cn lim an, (an), (bn), (cn) .
2.7.7. limxn + lim yn lim(xn + yn) lim(xn + yn) limxn + lim yn . limxn lim yn lim(xnyn) lim(xnyn) limxn lim yn xn > 0 yn > 0 . t > 0, lim(txn) = t limxn lim(txn) = t limxn. t < 0;
2.7.8. (yn) , lim(xn + yn) = limxn + lim yn lim(xn + yn) = limxn + lim yn. xn, yn > 0, (yn) , lim(xnyn) = limxn lim yn lim(xnyn) = limxn lim yn.
2.7.9. m N, (xn), (xn+m) .
2.7.10. (xnk) (xn). 2.5.14 (xnk) - (xn). limxn limxnk limxnk limxn.
2.7.11. [] x R (xn) > 0 xn Nx() n.[] X R (xn). (yn) X yn y R, y X . a < b < c d. (xn) [a, b) [c, d] [a, b] Q.
2.7.12. (xn) k xk limxn. (xn) k xk limxn.
2.7.13. [] limxn lim x1++xnn limx1++xn
n limxn. Cesro 2.3.41[];[] xn > 0 n, limxn lim n
x1 xn lim n
x1 xn
limxn.[] yn > 0 n y1+ +yn +, lim xnyn lim
x1++xny1++yn
lim x1++xny1++yn limxnyn.
67
-
2.7.14. 58 Cauchy (xn). x = limxn R x = limxn R. Cauchy (xn) , x, x R. x < x. l, u x < l < u < x. xm < l m xn > u