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.
Cauchy, Bolzano-Weierstrass, Convergence
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real-analysis
sequences-and-series
banach-spaces
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1In 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. – 2011-11-10
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0A 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. – 2011-11-10