If a subsequence of a Cauchy sequence converges, then the whole sequence converges.
Solution 1:
Very nice.
As an alternative, you could say that $d(x_{n_k},x)<\epsilon/2$ whenever $k>K.$ Then, you want to put $M=\max\{N,n_K\}.$ You know that $k>K$ if and only if $n_k>n_K,$ which allows you to draw the desired conclusion.