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  "body": "\n## Definition\n\n*Constructive analysis* is real and functional analysis developed within [[constructivism|F1P5SZ1XPH]], with intuitionistic logic.\nIn Bishop's approach, the basic notions are these.\n\n1. A *real number* is a sequence $(x_n)_{n \\ge 1}$ of rational numbers that is *regular*: $\\abs{x_m - x_n} \\le \\frac{1}{m} + \\frac{1}{n}$ for all $m, n \\ge 1$.\n2. Two real numbers $(x_n)$ and $(y_n)$ are *equal* if $\\abs{x_n - y_n} \\le \\frac{2}{n}$ for all $n \\ge 1$.\n3. A real number $(x_n)$ is *positive* if $x_n \u003e \\frac{1}{n}$ for some $n$, and $x \u003c y$ means that $y - x$ is positive.\n\nFunctions, continuity and other notions are then defined so that each existence statement comes with a method of construction.\n\n## Remarks\n\nA regular sequence is a [[Cauchy sequence|0QZMTN5AXX]] with an explicit modulus of convergence, and it defines a classical [[real number|08J2M1JYAG]].\nExample: the sequence $x_n = \\sum_{k=0}^{n} \\frac{1}{k!}$ is regular, since $\\abs{x_m - x_n} \\le \\sum_{k \u003e n} \\frac{1}{k!} \\le \\frac{1}{n}$ for $m \u003e n \\ge 1$; it defines $e$.\nNon-example: the statement \"for every real $x$, either $x = 0$ or $x \\ne 0$\" is not a theorem of constructive analysis.\nA proof would give a method to decide, for each regular sequence, whether all its terms stay within $\\frac{2}{n}$ of $0$.\nSuch a method would need the [[law of excluded middle|0RYYK8PG4S]] for statements about infinitely many terms.\nThe [[intermediate value theorem|00KAFBR32E]] holds in an approximate form: if $f$ is continuous on $[0, 1]$ with $f(0) \u003c 0 \u003c f(1)$, then for each $\\varepsilon \u003e 0$ there is $x$ with $\\abs{f(x)} \u003c \\varepsilon$.\n",
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