Convergent Sequence. Definition A sequence {p n } in a metric space (X, d) is said to converge if there is a point p X with the following property:

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1 Convergent Sequence Definition A sequence {p n } in a metric space (X, d) is said to converge if there is a point p X with the following property: ( ɛ > 0)( N)( n > N)d(p n, p) < ɛ In this case we also say that {p n } converges to p or that p is the limit of {p n } and we write p n p or lim n p n = p. If {p n } does not converge we say it diverges If there is any ambiguity we say {p n } converges/diverges in X The set of all p n is said to be the range of {p n } (which may be infinite or finite). We say {p n } is bounded if the range is bounded.

2 consider the following sequence of complex number (i.e. X = R 2 ) (a) If s n = 1/n then lim n s n = 0; the range is infinite, and the sequence is bounded. (b) If s n = n 2 then the sequence {s n } is divergent; the range is infinite, and the sequence is unbounded. (c) If s n = 1 + [( 1) n /n] then the sequence {s n } converges to 1, is bounded, and has infinite range. (d) If s n = i n the sequence {s n } is divergent, is bounded and has finite range. (e) If s n = 1 (n = 1, 2, 3,... ) then {s n } converges to 1is bounded and has finite range. Examples Notice that our definition of convergent depends not only on {p n } but also on X. For example {1/n : n N} converges in R 1 and diverges in (0, ).

3 Properties of Convergent Sequences (a) {p n } converges to p X if and only if every neighborhood of p contains p n for all but finitely many n. (b) If p, p X and if {p n } converges to p and to p then p = p (c) If {p n } converges then {p n } is bounded. (d) If E X and if p is a limit point of E, then there is a sequence {p n } in E such that p = lim n p n

4 Properties of Convergent Sequences Suppose {s n }, {t n } are complex sequence with lim n s n = s and lim n t n = t. Then (a) lim n (s n + t n ) = s + t (b) lim n c s n = c s and lim n c + s n = c + s for any number c. (c) lim n s n t n = st 1 (d) lim n s n = 1 s

5 Properties of Convergent Sequences (a) Suppose x n R k (n N) and x n = (α 1,n,..., α k,n ). Then {x n } converges to x = (α 1,..., α k ) if and only if lim α j,n = α j (1 j k) n (b) Suppose {x n }, {y n } are sequences in R k, {β n } is a sequence of real numbers, and x n x, y n y, β n β. Then lim (x n + y n n ) = x + y lim (x n y n n ) = x y lim β nx n = βx n

6 Definition Given a sequence {p n }, consider a sequence {n k } of positive integers such that n 1 < n 2 < n 3 <.... Then the sequence {p ni } is called a subsequence of {p n }. If {p ni } converges its limit is called a subsequential limit of {p n }. It is clear that {p n } converges to p if and only if every subsequence of {p n } converges to p.

7 and Compact Metric Spaces (a) If {p n } is a sequence in a compact metric space X, then some subsequence of {p n } converges to a point of X. (b) Every bounded sequence in R k contains a convergent subsequence.

8 Limits The subsequential limits of a sequence {p n } in a metric space X form a closed subset of X.

9 Cauchy Sequence Definition A sequence {p n } in a metric space (X, d) is said to be a Cauchy sequence if for every ɛ > 0 there is an integer N such that d(p n, p m ) < ɛ for all n, m N. Definition Let E be a nonempty subset of a metric space (X, d), and let S = {d(p, q) : p, q E}. The diameter of E is sup S. If {p n } is a sequence in X and if E N consists of the points p N, p N+1,..., it is clear that {p n } is a Cauchy sequence if and only if lim N diame N = 0

10 and Closed Sets (a) If E is the closure of a set E in a metric space X, then diam E = diam E (b) If K n is a sequence of compact sets in X such that K n K n+1 (n N) and if lim n diam K n = 0 then 1 K n consists of exactly one point.

11 and Convergent Sequences (a) In any metric space X, every convergent sequence is a Cauchy sequence. (b) If X is a compact metric space and if {p n } is a Cauchy sequence in X then {p n } converges to some point of X. (c) In R k every Cauchy sequence converges.

12 Complete Spaces Definition A metric space is said to be complete if every Cauchy sequence converges. Notice that all compact metric spaces are complete but there are metric spaces (like R k ) which are complete but not compact. Lemma Every closed subset of a complete metric space is complete.

13 Increasing/Decreasing Sequences Definition A sequence {s n } of real numbers is said to be (a) monotonically increasing if s n s n+1 for all n N (b) monotonically decreasing if s n s n+1 for all n N (c) monotonic if it is monotonically increasing or monotonically decreasing. Suppose {s n } is monotonic. Then {s n } converges if and only if {s n } is bounded.