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  "body": "\n## Summary\n\nThis construction defines the [[real numbers|08J2M1JYAG]] as limits of sequences of [[rational numbers|0NCJ5WXN6Z]].\nA sequence $(x_n)$ of rational numbers is a Cauchy sequence when, for each rational $\\varepsilon \u003e 0$, there is $N$ with $|x_m - x_n| \u003c \\varepsilon$ for all $m, n \\ge N$.\nTwo Cauchy sequences $(x_n)$ and $(y_n)$ are equivalent when $x_n - y_n$ tends to $0$: for each rational $\\varepsilon \u003e 0$, there is $N$ with $|x_n - y_n| \u003c \\varepsilon$ for all $n \\ge N$.\nA real number is an equivalence class of Cauchy sequences of rational numbers.\nAddition and multiplication act term by term: $[(x_n)] + [(y_n)] = [(x_n + y_n)]$ and $[(x_n)] \\cdot [(y_n)] = [(x_n y_n)]$.\n\n## Remarks\n\nThe rational number $q$ corresponds to the class of the constant sequence $(q, q, q, \\dots)$.\nThe sequence $1, 1.4, 1.41, 1.414, \\dots$ of decimal approximations represents $\\sqrt{2}$.\nThe result is a [[complete ordered field|0A9E6X0S3E]], with the [[order on the reals|0337CYB0VD]]. The same method completes any metric space.\n\n## Explanations\n\n### L1 human\n\nA real number is something that rational numbers can approximate as well as you like.\nThe construction takes the approximating sequences themselves as the new numbers.\nTwo sequences that get closer and closer to each other count as the same real number.\n",
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