REMARK. A sequence has at most one limit.

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1 CHAPTER 2 : SQUENCES IN R 2.1 LIMITS OF SEQUENCES DEFINITION. A sequence of real numbers {x n } is said to converge to a real number a R if and only if for every ǫ > 0 there is an N(ǫ) N such that n N(ǫ) implies x n a < ǫ. We will use the following notations interchangeably: (1) {x n } converges to a ; (2) x n converges to a ; (3) a = lim x n ; (4) x n a as n ; (5) the limit of {x n } exists and equals to a. Example. 1 n 0. Example. {( 1) n } does not have a limit REMARK. A sequence has at most one limit. DEFINITION. By a subsequence of a sequence {x n }, we shall mean a sequence of the form {x nk } k N where n k N and n 1 < n 2 < n 3 <.... Typeset by AMS-TEX 1

2 REMARK. If {x n } converges to a limit a amd {x nk } is any subsequence of {x n }, then x nk converges to a as k. DEFINITION. Let {x n } be a sequence of real numbers. (1) {x n } is said to be bounded above if and only if there is an M R such that x n M for all n N. (2) {x n } is said to be bounded below if and only if there is an m R such that x n m for all n N. (3) {x n } is said to be bounded if and only if it is bounded both below and above. Theorem. Every convergent sequence is bounded 2.2 LIMIT THEOREMS Theorem. (Squeeze Theorem) Suppose that {x n }, {y n } and {w n } are real sequences (1) If x n a and y n a as n and if there exists an N 0 N such that x n w n y n for n N 0, then w n a as n. (2) If x n 0 as n and {y n } is bounded, then x n y n 0 as n.

3 Example. Find lim 2 n cos(n 3 n 2 + n 13). Theorem. Let E R. If E has a finite supremum (respectively a finite infimum), then there is a sequence {x n } E such that x n sup E (respectively x n inf E) as n. Theorem. Suppose that {x n } and {y n } are real sequences and a R. If {x n } and {y n } are convergent, then (1) lim (x n + y n ) = lim x n + lim y n. (2) lim (ax n) = a( lim x n). (3) lim (x ny n ) = ( lim x n)( lim y n). (4) If y n 0 and lim y n 0 then lim x n y n = lim x n lim y n.

4 Example. lim n 3 +n n 3. Definition. Let {x n } be a sequence of real numbers. (1) {x n } is said to diverge to + (notation: x n + as n or lim x n = + ) if and only if for each M R there is an N N such that n N implies x n > M. (2) {x n } is said to diverge to (notation: x n as n or lim x n = ) if and only if for each M R there is an N N such that n N implies x n < M. Theorem. Suppose that {x n } and {y n } are real sequences such that lim x n = + (respectively lim x n = ). (1) If y n is bounded below (respectively, y n is bounded above), then lim (x n+y n ) = + (respectively, lim (x n+y n ) = ). (2) If a > 0, then lim (ax n) = + (respectively lim (ax n) = ). (3) If y n > M 0 for some M 0 > 0 and for all n N, then lim (x ny n ) = + (respectively, lim (x ny n ) = ).

5 (4) If y n is bounded and x n 0,then lim y n x n = 0. We will adopt the conventions (1) x + =, x =, x R. (2) x =, x ( ) =, x > 0. (3) x =, x ( ) =, x < 0. (4) + =, =. (5) = ( ) ( ) =. (6) ( ) = ( ) =. Corollary. Let {x n }, {y n } be real sequences and a, x, y be extended real numbers. If lim x n = x and lim y n = y. then lim (x n + y n ) = x + y (if x + y is not ), and lim ax n = ax, lim (x ny n ) = xy (if the right hand side is not 0 (± )).

6 Theorem. [Comparison theorem]. Suppose that {x n } ana {y n } are real sequences. If there is an N 0 N such that x n y n for all n N 0, then lim x n lim y n. In particular if x n [a, b] and lim x n = c then c [a, b]. 2.3 BOLZANO-WEIERSTRASS THEOREM DEFINITION. Let {x n } n N be a sequence of real numbers. (1) {x n } n N is said to be increasing (respectively, strictly increasing) if and only if x 1 x 2, (respectively, x 1 < x 2 < ). (2) {x n } n N is said to be decreasing (respectively, strictly decreasing) if and only if x 1 x 2,(respectively, x 1 > x 2 > ). (3) {x n } n N is said to be monotone if and only if it is either increasing or decreasing. Theorem. (Monotone Convergent Theorem) If {x n } n N is increasing and bounded above, or is decreasing and bounded below, then {x n } n N has a finite limit.

7 Example. If a < 1 then a n 0 as n. Example. If a > 0 then a 1 n 1 as n. DEFINITION. A sequence of sets {I n } n N is said to be nested if and only if I 1 I 2 Thorem. [Nested Intervals Property] If {I n } n N is a nested sequence of nonempty closed bounded intervals, then E = n N is nonempty. Moreover if the lenghts of these intervals satisfy I n 0 as, then E contains exactly one point. REMARK. The nested property might not hold if closed is omitted. REMARK. The nested property might not hold if bounded is omitted. Theorem. [Bolzano-Weierstrass Theorem] Every bounded sequence of real numbers has a convergent subsequence. I n

8 2.4 CAUCHY SEQUENCES DEFINITION. A sequence of points x n R is said to be Cauchy if and only if for every ǫ > 0 there is an N(ǫ) N such that n, m N(ǫ) imply x n x m < ǫ. Remark Cauchy. If {x n } is convergent, then {x n } is Theorem. [Cauchy] Let {x n } be a sequence of real numbers, then {x n } is Cauchy if and only if {x n } converges (to some point in R). Example Any real sequence {x n } that satisfies is convergent. x n x n+1 < 1 2 n, n N Remark A sequence that satisfies x n x n+1 0 is not necessarily Cauchy 2.5 LIMITS SUPREMUM AND INFIMUM

9 DEFINITION. Let {x n } be a real sequence. Then the limit supremum of {x n } is the extended real number lim sup x n := lim (sup x k ), k n and the limit infimum {x n } is the extended real number lim inf x n := lim ( inf k n x k). Let S n = {x n, } and let s n = sups n, t n = inf S n, then limsup x n = lim s n (decreasing limit) and liminf x n = lim t n (increasing limit). Remark. (1) If a < limsup x n, then there is a infinite subsequence x nk of x n such that a < x nk for all k. (2) If a > limsup x n then there is N(a) N such that a > x n for all n N(a). (3) If a > liminf x n, then there is a infinite subsequence x nk of x n such that a > x nk for all k. (4) If a < liminf x n then there is N(a) N such that a < x n for all n N(a).

10 Proof of (1). Since limsup x n is the decreasing limit of s n, we have a < s n for all n. We can apply the approximation property of supremum to s 1 and a to get x n1 such that a < x n1, and then do it again for s n1 anda to get x n2 and so on. Proof of (2). Since limsup x n is the decreasing limit of s n, there is N(a) such that s N(a) < a, then x n < a for all n > N(a). Example.x n = ( 1) n. Example.x n = n. Theorem. Let {x n } be a real squence, s = limsup, and t = liminf. Then there are subsequences {x nk } k N and {x lj } j N such that x nk s as k and x lj t as j. Theorem. Let {x n } be a real squence and x be an exrended real number. Then lim x n = x as n if and only if limsup = x = liminf x n. Theorem. Let {x n } be a real sequence. Then the limsup x n (respectively, liminf x n ) is the largest value (respectively, smallest value) to which some sub-

11 sequence of {x n } converges. Namely,if x nj j, then x as. lim sup x n x liminf x n Remark. If {x n } is any real sequence, then lim sup x n liminf x n. Remark. A real sequence {x n } is bounded above if and only if limsup x n <, and is bounded below if and only if liminf x n >. Theorem. If x n y n for large n, then lim sup x n limsup y n, and liminf x n liminf y n.