A sequence $\displaystyle (x_n)$ has a Cauchy subsequence if and only if it has a subsequence $\displaystyle (x_n_k)$ for which $\displaystyle d(x_n_k,x_n_{k+1}) < 2^{-k}$
A sequence $\displaystyle (x_n)$ has a Cauchy subsequence if and only if it has a subsequence $\displaystyle (x_n_k)$ for which $\displaystyle d(x_n_k,x_n_{k+1}) < 2^{-k}$
what have you tried, once again? This one is slightly tricky, the key thing to note is that $\displaystyle \displaystyle \sum2^{-k}=1$. So, use the triangle inequality, etc.