Theorem 1 (Weierstrass)

Let $f$ be a continuous real-valued function defined on a non-empty quasi-compact space $X$. Then there is at least one point $a \in X$ such that $f(a) = \sup_{x \in X} (f(x))$, and at least one point $b \in X$ such that $f(b) = \inf_{x \in X} f(x)$. For $f(X)$ is compact (Chapter I, § 9, no. 4, Theorem 2) and therefore closed in $\overline{\mathbf{R}}$; hence $f(X)$ contains its bounds.

This theorem is often stated in the form that a continuous real-valued function on a non-empty quasi-compact space attains its bounds.