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  "body": "\n## Statement\n\nLet $D \\subseteq \\mathbb{R}$ and let $f \\colon D \\to \\mathbb{R}$ be [[uniformly continuous|0Q4DR37DVE]].\nLet $(x_n)$ be a [[Cauchy sequence|0QZMTN5AXX]] in $D$.\nThen $(f(x_n))$ is a Cauchy sequence.\n\n## Remarks\n\nGiven $\\varepsilon \u003e 0$, take $\\delta$ from uniform continuity, and then $N$ with $|x_m - x_n| \u003c \\delta$ for $m, n \\ge N$.\nContinuity alone is not enough. The function $1/x$ on $(0, 1)$ maps the Cauchy sequence $1/n$ to the sequence $n$.\nMathlib states it as `UniformContinuous.comp_cauchySeq`.\n\n## Explanations\n\n### L1 human\n\nPoints of a Cauchy sequence crowd closer and closer together.\nA uniformly continuous function keeps their images crowding together too.\n",
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