Convergent sequences are Cauchy

Proposition. If $(x_n)$ is a convergent sequence in a metric space, then it is a Cauchy sequence.

Proof sketch. If $x_n\to a$, then for $\varepsilon>0$ choose $N$ such that $d(x_n,a)<\varepsilon/2$ for all $n\ge N$. For $m,n\ge N$,

by the triangle inequality.