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OpenStudy (anonymous):

prove a sequence \[(x_n)\] has a cauchy subsequence iff it has a subsequence \[(x_{n_k})\] such that \[d(x_{n_k},x_{n_{k+1}})<2^{−k} \] for all k

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OpenStudy (jamesj):

The (<=) direction of the proof should be easy: (xnk) is a Cauchy sequence. The (=>) direction is a bit tricker: Given a Cauchy subsequence, show that a subsequence of it must satisfy the 2^-k condition.

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