Compactness implies boundedness

Compactness implies boundedness: Let $(X,d)$ be a metric space and let $K\subseteq X$ be compact . Then $K$ is bounded : there exist $x_0\in X$ and $R>0$ such that $K\subseteq B(x_0,R).$

This is one of the basic “finiteness” consequences of compactness and is used to control sequences and covers.