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.

Proof sketch: Assume $f_n\uparrow f$ (the decreasing case is similar). If convergence were not uniform, there would exist $\varepsilon>0$ and points $x_n\in K$ with $f(x_n)-f_n(x_n)\ge\varepsilon$. By compactness, pass to a subsequence $x_{n_k}\to x$. Use continuity of $f$ and $f_{n_k}$ plus monotonicity in $n$ to show $f(x)-f_{n_k}(x)$ cannot stay $\ge\varepsilon$, a contradiction.