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  "body": "\n## Definition\n\nLet $S$ be a set and let $f_n \\colon S \\to \\mathbb{R}$ for $n \\ge 1$.\nThe sequence $(f_n)$ is uniformly Cauchy on $S$ if for every $\\varepsilon \u003e 0$ there is $N$ such that\n$$|f_n(x) - f_m(x)| \u003c \\varepsilon \\quad \\text{for all } m, n \\ge N \\text{ and all } x \\in S.$$\n\n## Remarks\n\nFor each $x$, the sequence $(f_n(x))$ is then a [[Cauchy sequence|0QZMTN5AXX]]. So it has a limit $f(x)$, by the [[Cauchy criterion|0ZA4AX4MW3]].\nA sequence of real functions is uniformly Cauchy on $S$ if and only if it [[converges uniformly|08HJRTG5N7]] on $S$.\nThis criterion proves uniform convergence without knowing the limit. Mathlib defines it as `UniformCauchySeqOn`.\n\n## Explanations\n\n### L1 human\n\nA uniformly Cauchy sequence has graphs that get close to each other, uniformly in $x$.\nSuch a sequence always converges uniformly to some function.\n",
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