Let A be a proper subset of R^m. A is compact, x in A, (x_n) sequence in A, every convergent subsequence of (x_n) converges to x.
(a) Prove the sequence (x_n) converges.
Is this because all the subsequences converge to the same limit?
(b) If A is not compact, show that (a) is not necessarily true.
If A is not compact, doesn't it imply that (x_n) doesn't necessarily have all subsequences as convergent?
Can you help?
Since A is compact in R^m, then A is bounded and closed. (x_n) is a sequence in A, then we know every bounded sequence must have a convergent subsequence. From the condition, each subsequence of (x_n) ...
A Compact Subset of R^m with Convergent Sequences is investigated. The solution is detailed and well presented. The response was given a rating of "5/5" by the student who originally posted the question.