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  "body": "\n## Statement\n\nLet $S \\subseteq \\mathbb{R}$, and let $f_n \\colon S \\to \\mathbb{R}$ be continuous for each $n$.\nIf $f_n \\to f$ [[uniformly on $S$|08HJRTG5N7]], then $f$ is continuous on $S$.\n\n## Remarks\n\nFix $c \\in S$ and $\\varepsilon \u003e 0$. Choose $n$ with $|f_n - f| \u003c \\varepsilon/3$ on $S$, and $\\delta$ for the [[continuity|0EEE2CZB2A]] of $f_n$ at $c$.\nThen $|f(x) - f(c)| \\le |f(x) - f_n(x)| + |f_n(x) - f_n(c)| + |f_n(c) - f(c)| \u003c \\varepsilon$ for $|x - c| \u003c \\delta$.\nThe [[sequence $x^n$ on $[0, 1]$|0YSRR1JTKZ]] shows that pointwise convergence is not enough. Mathlib states the theorem as `TendstoUniformly.continuous`.\n\n## History\n\nAugustin-Louis Cauchy claimed in 1821 that a series of continuous functions has a continuous sum, whenever it converges.\nIn 1826, Niels Henrik Abel found counterexamples among Fourier series. In modern terms, Cauchy's argument needs uniform convergence.\n\n## Explanations\n\n### L1 human\n\nIf continuous graphs approach a limit graph uniformly, the limit graph has no jumps.\nA jump in the limit would force some $f_n$ to jump as well.\n",
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  "name": "Uniform limit theorem",
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