Dini's theorem

Statement

Let $a < b$, and let $f_n \colon [a, b] \to \mathbb{R}$ be continuous functions with $f_n(x) \le f_{n+1}(x)$ for all $n$ and all $x$. Assume that $(f_n)$ converges pointwise to a continuous function $f$ on $[a, b]$. Then $(f_n)$ converges uniformly to $f$ on $[a, b]$.

Remarks

The same holds for decreasing sequences, and for any compact space in place of $[a, b]$. One proof uses the Bolzano–Weierstrass theorem. Another uses the open sets $\{x : f(x) - f_n(x) < \varepsilon\}$, which increase and cover $[a, b]$. All hypotheses are needed. The sequence $x^n$ on $[0, 1]$ is monotone, but its limit is not continuous. Mathlib states a version as Monotone.tendstoLocallyUniformly_of_forall_tendsto.

History

The theorem is named after Ulisse Dini. He gave the first version in an 1878 book on real functions, printed in Pisa.

Explanations

L1 human

If continuous functions increase to a continuous limit on a closed bounded interval, they get close to it everywhere at once. Monotonicity and compactness rule out the bad behavior of examples such as $x^n$.