Basic Analysis I — Real Analysis · Problem 7.3.13

Metric Spaces — Jiri Lebl

Mathematics — Real Analysis · sequences-and-convergence

Problem

Let (X,d)(X, d) be a metric space and {xn}n=1∞\{x_n\}_{n=1}^\infty a sequence in XX. Prove that {xn}n=1∞\{x_n\}_{n=1}^\infty converges to p∈Xp \in X if and only if every subsequence of {xn}n=1∞\{x_n\}_{n=1}^\infty has a subsequence that converges to pp.

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