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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 $n \\ge 0$.\nIf the [[series|0MRE6XRE5V]] $\\sum_{n=0}^{\\infty} f_n$ converges uniformly on $S$, then its sum is continuous on $S$.\n\n## Remarks\n\nThe partial sums are continuous, and they converge uniformly. Apply the [[uniform limit theorem|0X9XT05JVC]].\nUniform convergence often comes from the [[M-test|0NX5NWAFC9]]. Mathlib states the result in this form as `continuous_tsum`.\nFor example, $\\sum_{n=1}^{\\infty} \\frac{\\cos(n x)}{n^2}$ is continuous on $\\mathbb{R}$.\n\n## Explanations\n\n### L1 human\n\nAdding infinitely many continuous functions gives a continuous function, if the sum converges uniformly.\n",
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