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  "body": "\n## Statement\n\nLet $a \\le b$ and let $f \\colon [a, b] \\to \\mathbb{R}$ be [[continuous|0P27W9RW0G]].\nThen $f$ is [[uniformly continuous|0A3CK6DVVB]] on $[a, b]$.\n\n## Remarks\n\nSuppose not. Then there is $\\varepsilon \u003e 0$ and points $x_n, y_n$ with $|x_n - y_n| \u003c 1/n$ and $|f(x_n) - f(y_n)| \\ge \\varepsilon$. A [[convergent subsequence|0KDJD249DT]] of $(x_n)$ leads to a contradiction with continuity.\nThe interval must be closed and bounded. The function $1/x$ is continuous on $(0, 1]$ but not uniformly continuous there.\nThe name honors Eduard Heine and Georg Cantor. Mathlib states it for compact sets, as `IsCompact.uniformContinuousOn_of_continuous`.\n\n## Explanations\n\n### L1 human\n\nOn a closed bounded interval, continuity at each point upgrades to uniform continuity.\nThere is no room for the function to become steeper and steeper.\n",
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