Cauchy sequence is bounded

Cauchy sequence is bounded: Let $(X,d)$ be a metric space and let $(x_n)$ be a Cauchy sequence in $X$. Then $\{x_n:n\in\mathbb{N}\}$ is a bounded set ; equivalently, there exist $x_0\in X$ and $R>0$ such that $d(x_n,x_0)\le R$ for all $n$.

This is often paired with completeness results: Cauchy behavior already forces the sequence to stay inside some closed ball .