Cauchy sequences and subsequences

I have the following theorem

If $\displaystyle {x_n}$ is Cauchy and $\displaystyle {x_n_{k}}$ is a convergent subsequence, then $\displaystyle {x_n}$ converges.

$\displaystyle n,m > N$ implies $\displaystyle d(x_n,x_m) < \frac{\epsilon}{2}$

and

$\displaystyle d(x_n_{k},x) < \frac{\epsilon}{2}$

I want to say

By triangle inequality, $\displaystyle d(x_n,x) <= d(x_n,x_m) + d(x_n_{k},x) < \epsilon$

But I am not really sure if the triangle inequality applies here. Is this correct or can I make a different argument?

Thank you very much.