Assignment:
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?
Provide complete and step by step solution for the question and show calculations and use formulas.