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  "body": "\n## Statement\n\nLet $\\alpha$ be a real number and let $N$ be a positive integer.\nThen there are integers $p$ and $q$ with $1 \\le q \\le N$ and\n$$|q \\alpha - p| \\le \\frac{1}{N + 1}.$$\n\n## Remarks\n\nA corollary: if $\\alpha$ is irrational, then $\\left| \\alpha - \\frac{p}{q} \\right| \u003c \\frac{1}{q^2}$ for infinitely many fractions $p / q$.\nThe proof uses the pigeonhole principle. Among the $N + 1$ fractional parts of $0, \\alpha, 2\\alpha, \\dots, N\\alpha$, two lie in one of $N + 1$ intervals of length $\\frac{1}{N + 1}$.\nThe [[convergents|0E70M92FE3]] of the continued fraction of $\\alpha$ give such fractions explicitly.\n\n## Explanations\n\n### L1 human\n\nEach real number has fractions $p/q$ very close to it, with an error much smaller than $1/q$.\nThe theorem gives the bound $1/q^2$ for infinitely many denominators when $\\alpha$ is irrational.\n",
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