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  "body": "\n## Definition\n\nLet $\\mathcal{U}$ be a nonprincipal [[ultrafilter|J64Y3JC0RE]] on $\\N$.\nFor [[sequences|0G9EKM0E15]] $a = (a_n)$ and $b = (b_n)$ of [[real numbers|08J2M1JYAG]], put $a \\sim b$ if $\\set{n \\in \\N : a_n = b_n} \\in \\mathcal{U}$.\nThis is an [[equivalence relation|0G8VEZ3WTK]]; write $[a]$ for the class of $a$.\nThe set of *hyperreal numbers* is ${}^*\\R = \\R^{\\N} / {\\sim}$, with the following structure.\n\n1. $[a] + [b] = [(a_n + b_n)]$ and $[a] \\cdot [b] = [(a_n b_n)]$.\n2. $[a] \u003c [b]$ if $\\set{n \\in \\N : a_n \u003c b_n} \\in \\mathcal{U}$.\n3. Each real number $r$ is identified with the class of the constant sequence $(r, r, r, \\dots)$.\n\nA hyperreal $x$ is *infinitesimal* if $\\abs{x} \u003c r$ for every real $r \u003e 0$, and *infinite* if $\\abs{x} \u003e r$ for every real $r$.\n\n## Remarks\n\nThe operations and the order are well defined, and ${}^*\\R$ is an [[ordered field|0P3VCBDEE2]] containing $\\R$; it is the [[ultrapower|AX9Q86GB8M]] of $\\R$ by $\\mathcal{U}$.\n\nExample: $\\varepsilon = [(1/(n+1))]$ is a positive infinitesimal.\nFor a real $r \u003e 0$, the set $\\set{n : 1/(n+1) \u003c r}$ is cofinite, and a nonprincipal ultrafilter contains every cofinite set, so $\\varepsilon \u003c r$.\nAlso $\\varepsilon \u003e 0$ because every term is positive.\nSo ${}^*\\R$ is not [[Archimedean|0XH79S06J3]].\n\nNon-example: if $\\mathcal{U}$ is the principal ultrafilter of the sets containing a fixed $k \\in \\N$, then $[a] \\mapsto a_k$ is an isomorphism onto $\\R$, and there are no nonzero infinitesimals.\n\nUnder the continuum hypothesis, the ordered field ${}^*\\R$ does not depend on the choice of $\\mathcal{U}$, up to isomorphism.\n",
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