# Thread: Cauchy Sequence in Metric Space

1. ## Cauchy Sequence in Metric Space

Let $\{x_n\}$ be a Cauchy sequence in $X$. Show that there exists a subsequence $\{x_{n_{k}}\}$ of $\{x_n\}$ such that $\sum_{k=1}^{\infty} d(x_{n_{k}},x_{n_{k+1}}) < \infty$.

I can only say that since $\{x_n\}$ is a Cauchy sequence, $\{x_n\}$ is bounded, and since $\{x_n\}$ is bounded, by Bolzano-Weierstrass Theorem, the sequence has a convergent subsequence. So how do I go about continuing from here?

You can construct by induction the sequence $\{n_k\}$ such that $d(x_{n_k},x_{n_{k+1}})\leq 2^{-k}$.
Note that we cannot necessary extract a converging subsequence of a bounded sequence, for example if $X$ is an infinite dimensional vector space.