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  "body": "\n## Statement\n\nLet $a \u003c b$, 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 $x$.\nAssume that $(f_n)$ converges [[pointwise|056FSKG32J]] to a continuous function $f$ on $[a, b]$.\nThen $(f_n)$ converges [[uniformly|08FPABDQ4C]] to $f$ on $[a, b]$.\n\n## Remarks\n\nThe same holds for decreasing sequences, and for any compact space in place of $[a, b]$.\nOne proof uses the [[Bolzano–Weierstrass theorem|0KDJD249DT]]. Another uses the open sets $\\{x : f(x) - f_n(x) \u003c \\varepsilon\\}$, which increase and cover $[a, b]$.\nAll 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`.\n\n## History\n\nThe theorem is named after Ulisse Dini. He gave the first version in an 1878 book on real functions, printed in Pisa.\n\n## Explanations\n\n### L1 human\n\nIf continuous functions increase to a continuous limit on a closed bounded interval, they get close to it everywhere at once.\nMonotonicity and compactness rule out the bad behavior of examples such as $x^n$.\n",
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