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Dirichlet's Approximation Theorem
"If \(\alpha\) is a real number and \(n\) is a positive integer, then there exist integers \(a\) and \(b\) with \(1\leq a\leq n\) such that \(|a\alpha-b|<1/n\)."
Proof. Consider the \(n+1\) numbers \(0,\{\alpha\},\{2\alpha\}\), ..., \(\{n\alpha\}\). These \(n+1\) numbers are the fractional parts of the numbers \(j\alpha\), \(j=0,1\), ..., \(n\), so that \(0\leq\{j\alpha\}<1\) for \(j=0,1\), ..., \(n\). Each of these \(n+1\) numbers lies in one of the \(n\) disjoint intervals \(0\leq x<1/n\), \(1\leq x<2/n\), ..., \((j-1)\leq x

Sabes algo de eso jose?

si