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  "body": "\n## Statement\n\nLet $x$ be an irrational real number, and let $p_n / q_n$ be the [[convergents|0SRWNR42BQ]] of its [[continued fraction expansion|0SJ7Z6VKWK]].\nThen\n$$\\left| x - \\frac{p_n}{q_n} \\right| \u003c \\frac{1}{q_n q_{n+1}} \\le \\frac{1}{q_n^2}$$\nfor each $n \\ge 0$, and $p_n / q_n \\to x$ as $n \\to \\infty$.\n\n## Remarks\n\nThe convergents with even $n$ increase toward $x$, and the convergents with odd $n$ decrease toward $x$.\nFor a rational $x$, the last convergent is equal to $x$.\nConversely, each infinite simple continued fraction converges to an irrational number, and this number has that continued fraction as its expansion.\nFor example, $\\frac{355}{113}$ is a convergent of $\\pi$, and $\\left| \\pi - \\frac{355}{113} \\right| \u003c 3 \\cdot 10^{-7}$.\n\n## Explanations\n\n### L1 human\n\nThe convergents of $x$ get closer and closer to $x$, and the error is smaller than one over the square of the denominator.\nThis makes them very good rational approximations of $x$.\n",
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  "name": "Convergents converge to the number",
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