Continuous functions on compact sets are bounded

Let $(X,d)$ be a metric space , let $K\subseteq X$ be compact , and let $f:K\to\mathbb{R}$ be continuous .

Proposition: The function $f$ is bounded on $K$: there exists $M<\infty$ such that $|f(x)|\le M$ for all $x\in K$.

This proposition is one of the core reasons compactness is the right hypothesis for global control from local continuity.