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I am having trouble understanding the theorem below, so I will post it here as a reminder to myself to review it as often as I can. Also, I may get lucky enough to have someone clarify the questions that I will most likely post here as I read through it time and time again.

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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
Your question prompted me to search this topic and try and learn about it. During my google quest, I came across the original German paper on this topic and thought it might interest you as I /think/ you are a German speaker. http://bibliothek.bbaw.de/bibliothek-digital/digitalequellen/schriften/anzeige?band=07-abh/1837&seite:int=00000286
Sabes algo de eso jose?

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