Cauchy implies bounded

Cauchy implies bounded: Let $(X,d)$ be a metric space and let $(x_n)$ be a Cauchy sequence in $X$. Then $(x_n)$ is bounded : there exist $x\in X$ and $R>0$ such that $d(x_n,x)\le R$ for all $n$.

This is a routine but important tool: Cauchy behavior already forces global control on the sequence.