Dini's Theorem

On a compact space, monotone pointwise convergence of continuous functions to a continuous limit is uniform

Dini’s Theorem

Dini’s Theorem: Let $K$ be a compact metric space and let $f_n:K\to\mathbb{R}$ be continuous for all $n$ . Suppose:

for each $x\in K$ , the sequence $f_n(x)$ is monotone in $n$ (either nondecreasing for all $x$ , or nonincreasing for all $x$ ), and
$f_n(x)\to f(x)$ pointwise on $K$ for some continuous function $f:K\to\mathbb{R}$ .

Then $f_n\to f$ uniformly on $K$ .

Dini’s theorem is a compactness-based upgrade from pointwise to uniform convergence when monotonicity is present. It is especially useful when uniform estimates are hard but monotone structure is available.