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This theorem is a consequence of the [[pigeonhole principle]]. [[Peter Gustav Lejeune Dirichlet]] who proved the result used the same principle in other contexts (for example, the [[Pell equation]]) and by naming the principle (in German) popularized its use, though its status in textbook terms comes later.<ref>http://jeff560.tripod.com/p.html for a number of historical references.</ref> The method extends to simultaneous approximation.<ref>{{Springer|id=d/d032940|title=Dirichlet theorem}}</ref>
'''Proof outline''': Let <math>\alpha</math> be an irrational number and <math>
One can divide the interval <math>[0, 1)</math> into <math>N</math> smaller intervals of measure <math>\frac{1}{N}</math>. Now, we have <math>N+1</math> numbers <math>x_0,x_1,...,x_N</math> and <math>N</math> intervals. Therefore, by the pigeonhole principle, at least two of them are in the same interval. We can call those <math>x_i,x_j</math> such that <math>i < j</math>. Now:
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