Question

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

Answer 1

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.