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  "body": "\n## Statement\n\nLet $(a_n)$ be a real sequence.\nThen $(a_n)$ converges if and only if $(a_n)$ is a [[Cauchy sequence|0QZMTN5AXX]].\n\n## Remarks\n\nOne direction is [[convergent sequences are Cauchy|0QY87R9CY3]].\nFor the other direction, a Cauchy sequence is [[bounded|0G6D8E7CTA]], so it has a convergent subsequence by the [[Bolzano–Weierstrass theorem|0KDJD249DT]]. A Cauchy sequence with a convergent subsequence converges to the same limit.\nThe criterion says that $\\mathbb{R}$ is a complete metric space. It fails in $\\mathbb{Q}$.\nIts value is that it proves convergence without a candidate for the limit.\n\n## Explanations\n\n### L1 human\n\nTo show that a sequence has a limit, you often need to know the limit first.\nThe Cauchy criterion avoids this. It is enough to show that the terms become close to each other.\n",
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