Dini's theorem

Dini’s theorem: Let $K$ be a compact topological space and let $f_n:K\to\mathbb{R}$ be continuous for every $n$. Assume $(f_n)$ is monotone in $n$ (either $f_n(x)\le f_{n+1}(x)$ for all $x\in K$, or $f_n(x)\ge f_{n+1}(x)$ for all $x\in K$) and that $f_n\to f$ pointwise on $K$, where $f$ is continuous. Then $f_n\to f$ uniformly on $K$.

This is a compactness-based upgrade from pointwise convergence to uniform convergence under the additional hypothesis of monotonicity, as in a monotone sequence of functions .