Basic Analysis I — Real Analysis · Problem 7.4.20

Metric Spaces — Jiri Lebl

Mathematics — Real Analysis · completeness-and-compactness

Problem

Let (X,d)(X, d) be a complete metric space. We say a set S⊂XS \subset X is relatively compact if the closure S‾\overline{S} is compact. Prove that S⊂XS \subset X is relatively compact if and only if given any sequence {xn}n=1∞\{x_n\}_{n=1}^\infty in SS, there exists a subsequence {xnk}k=1∞\{x_{n_k}\}_{k=1}^\infty that converges (in XX).

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