# Cauchy, Bolzano-Weierstrass, Convergence

Why is it that if for every bounded sequence we can find a convergent subsequence (in a normed vector space) then every Cauchy sequence converges (in this normed space)? Thanks.

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In the metric generated by the nomr ||.|| , i.e., d(x,y):=||x-y||, compactness is equivalent to every sequence having a convergent subsequence. And a compact metric space is complete. – gary Nov 10 '11 at 20:01
A Cauchy sequence is bounded, then continue as in the case of the real numbers: if a Cauchy sequence has a convergent subsequence, then the whole sequence converges. – GEdgar Nov 10 '11 at 20:02

Exercise 2: If a Cauchy sequence $\{x_n\}$ has a convergent subsequence $x_{n_k} \to x$, then $x_n \to x$.