Dini's theorem - Knowlpedia

Empty set and whole space: ∅ , X ∈ T \varnothing,X\in\mathcal T ∅ , X ∈ T .
Arbitrary unions: the union of any collection of members of T \mathcal T T belongs to T \mathcal T T .
Finite intersections: the intersection of finitely many members of T \mathcal T T belongs to T \mathcal T T .

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 $K$ .

Remarks

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.