Compactness implies boundedness

Compactness implies boundedness: Let $(X,d)$ be a metric space and let $K\subseteq X$ be a compact set . Then $K$ is a bounded set ; equivalently, $\operatorname{diam}(K)<\infty$, where $\operatorname{diam}$ is the diameter induced by $d$.

This is a first step toward “compact sets behave like finite sets” in metric settings, and it complements compactness implies total boundedness .