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  "body": "\n## Statement\n\nThe ordered field $\\mathbb{R}$ of [[real numbers|08J2M1JYAG]] is an [[Archimedean ordered field|0HQARMKEV5]].\nThat is, for all real numbers $x$ and $y$ with $y \u003e 0$, there is a natural number $n$ with $n y \u003e x$.\n\n## Remarks\n\nThe proof uses the [[completeness axiom|0AXWET80YW]]. If $\\mathbb{N}$ were bounded above, let $b = \\sup \\mathbb{N}$. Then $b - 1$ is not an upper bound, so $n \u003e b - 1$ for some $n$, and $n + 1 \u003e b$, a contradiction.\nThus each [[complete ordered field|0A9E6X0S3E]] is Archimedean.\n\n## Explanations\n\n### L1 human\n\nHowever small your step and however far your goal on the real line, finitely many steps get you past the goal.\nThere are no infinitely large real numbers.\n",
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