Continuous function on a compact set is bounded

Corollary: Let $(X,d)$ be a compact metric space and let $f:X\to\mathbb{R}$ be continuous . Then $f$ is bounded : there exists $M<\infty$ such that $|f(x)|\le M$ for all $x\in X$.

Connection to parent theorem: By the extreme value theorem , $f$ attains a maximum $M_+$ and a minimum $M_-$. Then $|f(x)|\le \max\{|M_+|,|M_-|\}$.