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  "body": "\n## Statement\n\nLet $(a_n)$ be a [[Cauchy sequence|0QZMTN5AXX]] of real numbers.\nThen $(a_n)$ is [[bounded|05FQ3VGAR9]].\n\n## Remarks\n\nTake $\\varepsilon = 1$ to get $N$ with $|a_n - a_N| \u003c 1$ for all $n \\ge N$. Then $|a_n| \\le \\max(|a_0|, \\dots, |a_{N-1}|, |a_N| + 1)$ for all $n$.\nThe statement holds in each metric space.\nThe proof of the [[Cauchy criterion|0ZA4AX4MW3]] uses this fact.\n\n## Explanations\n\n### L1 human\n\nFrom some index on, all terms are within distance $1$ of one fixed term.\nThe finitely many terms before that index are bounded too, so the whole sequence is bounded.\n",
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