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

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

Thanks in advance.