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  "body": "\n## Statement\n\nLet $a \u003c b$, and let $f_n \\colon [a, b] \\to \\mathbb{R}$ be Riemann integrable for each $n$.\nIf $f_n \\to f$ [[uniformly|08FPABDQ4C]] on $[a, b]$, then $f$ is Riemann integrable on $[a, b]$, and\n$$\\lim_{n \\to \\infty} \\int_a^b f_n(x) \\, dx = \\int_a^b f(x) \\, dx.$$\n\n## Remarks\n\nLet $\\varepsilon_n = \\sup |f_n - f|$. Then $f_n - \\varepsilon_n \\le f \\le f_n + \\varepsilon_n$. So the upper and lower sums of $f$ differ by at most $2 \\varepsilon_n (b - a)$ more than those of $f_n$. This gives integrability, by [[Riemann's criterion|017DH8WXTE]].\nBy [[monotonicity|0MBWQKM8QR]], $\\big| \\int_a^b f_n - \\int_a^b f \\big| \\le \\varepsilon_n (b - a) \\to 0$.\nPointwise convergence is not enough. On $[0, 1]$, the functions $f_n(x) = n^2 x (1 - x^2)^n$ tend to $0$ pointwise, but $\\int_0^1 f_n = \\frac{n^2}{2n + 2} \\to \\infty$.\n\n## Explanations\n\n### L1 human\n\nIf $f_n$ is uniformly close to $f$, the areas under their graphs are close too.\nSo we may take the limit inside the integral.\n",
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  "name": "Uniform limits commute with integration",
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