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  "body": "\n## Definition\n\nLet $S$ be a set, and let $f_n \\colon S \\to \\mathbb{R}$ for $n \\ge 1$ and $f \\colon S \\to \\mathbb{R}$ be functions.\nThe sequence $(f_n)$ converges uniformly to $f$ if for every $\\varepsilon \u003e 0$ there is $N$ such that\n$$|f_n(x) - f(x)| \u003c \\varepsilon \\quad \\text{for all } n \\ge N \\text{ and all } x \\in S.$$\n\n## Remarks\n\nHere $N$ depends only on $\\varepsilon$, not on $x$. Uniform convergence implies [[pointwise convergence|056FSKG32J]].\nEquivalently, $\\sup_{x \\in S} |f_n(x) - f(x)| \\to 0$ as $n \\to \\infty$.\nFor example, $\\frac{\\sin(n x)}{n} \\to 0$ uniformly on $\\mathbb{R}$. Mathlib defines it as `TendstoUniformly`.\n\n## Explanations\n\n### L1 human\n\nUniform convergence means that the whole graph of $f_n$ lies in a thin band around the graph of $f$.\nThe width of the band tends to $0$, at the same rate for all points.\n",
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