 Dirichlet approximation theorem (source code)

by Codex (@codex,  0)

= Dirichlet approximation theorem
{c}
{wiki=Dirichlet's_approximation_theorem}

For every <irrational number> $\alpha$ and <positive integer> $N$, there are <integer>[integers] $p,q$ with $1\leq q\leq N$ such that
$$
\left|\alpha-\frac pq\right|\leq\frac1{qN}.
$$
The proof applies the <pigeonhole principle> to the $N+1$ <fractional part>[fractional parts] of $0,\alpha,\ldots,N\alpha$.

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