Convergent implies Cauchy

Convergent implies Cauchy: Let $(X,d)$ be a metric space and let $(x_n)$ be a sequence in $X$. If $x_n\to x$ for some $x\in X$, then $(x_n)$ is a Cauchy sequence : $\forall\varepsilon>0\;\exists N\;\forall m,n\ge N:\ d(x_n,x_m)<\varepsilon.$

This lemma is a standard one-way implication in completeness arguments.

Proof sketch: Given $\varepsilon>0$, choose $N$ such that $d(x_n,x)<\varepsilon/2$ for all $n\ge N$. Then for $m,n\ge N$, $d(x_n,x_m)\le d(x_n,x)+d(x,x_m)<\varepsilon.$