Cauchy sequence - trick question?

Question

The question is the following:

Show that if a subsequence $\{x_{n_k} \}$ of a Cauchy sequence $\{ x_n\}$ is convergent, then $\{x_n\}$ is convergent.

I thought that all Cauchy sequences are convergent. At least in $\mathbb{R}$?

Answer 1

One way of proving that every Cauchy sequence (in $\mathbb R$) converges is to follow these steps:

1) Prove every Cauchy sequence is bounded

2) Use Bolzano-Weierstrass to prove every Cauchy sequence thus has a convergent subsequence

3) Prove that if a Cauchy sequence has at least one convergent subsequence, then the full sequence converges to the same limit.

You're just being asked to prove the last step.