Extreme Value Theorem

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

This theorem is a cornerstone of analysis and optimization: compactness is exactly the hypothesis ensuring that suprema /infima are achieved.

Proof sketch (optional): The image $f(X)$ is compact in $\mathbb{R}$ because $f$ is continuous and $X$ is compact. Compact subsets of $\mathbb{R}$ are closed and bounded , so $f(X)$ has a maximum and minimum; pull back the points in $X$ realizing those values.