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  "body": "\n## Statement\n\nLet $a \u003c b$, and let $f_n \\colon [a, b] \\to \\mathbb{R}$ be differentiable for each $n$.\nAssume that $(f_n(x_0))$ converges for some $x_0 \\in [a, b]$, and that $(f_n')$ converges [[uniformly|08FPABDQ4C]] on $[a, b]$ to a function $g$.\nThen $(f_n)$ converges uniformly on $[a, b]$ to a differentiable function $f$, and $f' = g$.\n\n## Remarks\n\nThe [[mean value theorem|0BARQY52N5]] applied to $f_n - f_m$ shows that $(f_n)$ is [[uniformly Cauchy|01XAY17QA3]].\nUniform convergence of $f_n$ alone is not enough. The functions $\\frac{\\sin(n^2 x)}{n}$ tend to $0$ uniformly, but their derivatives $n \\cos(n^2 x)$ do not converge.\nMathlib states a version as `hasDerivAt_of_tendstoUniformly`.\n\n## Explanations\n\n### L1 human\n\nDerivatives are sensitive to small wiggles, so uniform convergence of functions does not control them.\nWe need uniform convergence of the derivatives themselves.\n",
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  "msc": [
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  "name": "Uniform convergence of derivatives allows term-by-term differentiation",
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