A property of a Cauchy sequence

Question

Given an injective Cauchy sequence $(x_n)_n$ prove that it has a subsequence $(y_m)_m$ such that $$ d(y_{m+1},y_{n+1}) < d(y_m,y_n)/2 $$ for all $m,n \in \mathbb{N}$. I tried to prove this via induction, namely if we have taken the first n terms we take $\epsilon = \min(d(y_i,y_j) /2$ and work from their but it didn't work out.