Continuous function attains max and min on a compact set

Corollary (Extreme Value Theorem): Let $(X,d)$ be a compact metric space and let $f:X\to\mathbb{R}$ be continuous . Then $f$ attains its maximum and minimum : there exist $x_{\min},x_{\max}\in X$ such that $f(x_{\min})=\min_{x\in X} f(x),\qquad f(x_{\max})=\max_{x\in X} f(x).$

Connection to parent theorem: This is the extreme value theorem ; it follows because $f(X)$ is compact in $\mathbb{R}$, hence has a minimum and maximum.