You can construct by induction the sequence such that .
Note that we cannot necessary extract a converging subsequence of a bounded sequence, for example if is an infinite dimensional vector space.
Let be a Cauchy sequence in . Show that there exists a subsequence of such that .
I can only say that since is a Cauchy sequence, is bounded, and since is bounded, by Bolzano-Weierstrass Theorem, the sequence has a convergent subsequence. So how do I go about continuing from here?
Thanks in advance.