Semicontinuous Functions Attain Their Extrema on a Compact Set

Let $(X,d)$ be a metric space, equipped with the collection of all subsets that are open in $(X,d)$, which is a topology by Metric Open Sets Form a Topology. Let $K\subseteq X$ be nonempty and compact in $X$. Let $\mathbb{R}$ be the set of real numbers with the order $\le$ of its ordered field structure. Then the following hold.

1. (Maximum) If $u:K\to\mathbb{R}$ is upper semicontinuous on $K$, then there exists $x_{\max}\in K$ such that $u(x)\le u(x_{\max})$ for every $x\in K$.

2. (Minimum) If $w:K\to\mathbb{R}$ is lower semicontinuous on $K$, then there exists $x_{\min}\in K$ such that $w(x_{\min})\le w(x)$ for every $x\in K$.