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  "body": "\n## Definition\n\nConstruct the real numbers as classes of Cauchy sequences of rational numbers, as in the [[construction from Cauchy sequences|0D8HE681SC]].\nLet $x$ and $y$ be real numbers, represented by the Cauchy sequences $(x_n)$ and $(y_n)$ of rational numbers.\nThe notation $x \u003c y$ means that there are a rational number $\\varepsilon \u003e 0$ and a natural number $N$ such that $y_n - x_n \u003e \\varepsilon$ for each $n \\ge N$.\nThe notation $x \\le y$ means that $x \u003c y$ or $x = y$.\n\n## Remarks\n\nThe relation $x \u003c y$ does not depend on the choice of the sequences that represent $x$ and $y$.\nFor all real numbers $x$ and $y$, exactly one of $x \u003c y$, $x = y$ and $y \u003c x$ holds. Thus $\\le$ is a total order.\nOn the rational numbers, this order agrees with the [[order of the rationals|09RFKMBQ63]].\nWith [[Dedekind cuts|04QNZZ4HHY]], the order is inclusion of the lower sets of the cuts.\n\n## Explanations\n\n### L1 human\n\nTwo real numbers are compared through their rational approximations.\nSuppose that, from some point on, the approximations of $y$ exceed those of $x$ by a fixed positive gap.\nThen $x$ is less than $y$.\nFor example, $1.41, 1.414, 1.4142, \\dots$ approximates $\\sqrt{2}$, and these approximations are larger than $1.4$ by more than $0.005$, thus $1.4 \u003c \\sqrt{2}$.\n",
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