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.

Proof sketch: The image $f(K)$ is compact in $\mathbb{R}$ because $f$ is continuous and $K$ is compact. Compact subsets of $\mathbb{R}$ are bounded, so $f(K)$ is bounded. Equivalently, apply the extreme value theorem to $f$ and $-f$.