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  "body": "\n## Statement\n\nLet $(X, d)$ be a [[metric space|063W5JMY85]] and let $(x_n)_{n \\ge 1}$ be a sequence in $X$.\nFor each $N \\ge 1$, let $T_N = \\{x_n : n \\ge N\\}$.\nThen the following two conditions are equivalent:\n\n1. For each real $\\varepsilon \u003e 0$ there is an integer $N$ such that $d(x_m, x_n) \u003c \\varepsilon$ for all $m, n \\ge N$.\n2. The [[diameters|0692CGQF00]] $\\operatorname{diam}(T_N)$ tend to $0$ as $N \\to \\infty$.\n\nThe sequence $(x_n)$ is a Cauchy sequence when it satisfies these conditions.\n\n## Remarks\n\nCondition 1 is the usual definition of a Cauchy sequence.\nEach convergent sequence is a Cauchy sequence: if $x_n \\to x$, then $d(x_m, x_n) \\le d(x_m, x) + d(x, x_n)$.\nIn $\\mathbb{Q}$ with the usual metric, the decimal approximations $1, 1.4, 1.41, 1.414, \\ldots$ of $\\sqrt{2}$ form a Cauchy sequence that does not converge in $\\mathbb{Q}$.\nEach Cauchy sequence is [[bounded|0P953MG05E]].\n\n## Explanations\n\n### L1 human\n\nThe terms of a Cauchy sequence get closer and closer to each other.\nYou can check this without knowing a limit, and this is the point. A Cauchy sequence looks as if it converges. But the space can have no point for it to converge to.\n",
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