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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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
1 Answers
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Exercise 1: Every Cauchy sequence is bounded.
Exercise 2: If a Cauchy sequence $\{x_n\}$ has a convergent subsequence $x_{n_k} \to x$, then $x_n \to x$.
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0Thanks! Silly me didn't notice the cauchy v.s. boundedness link! – 2011-11-10