Approximation of Irrationals

If is irrational and is any positive integer, there is a fraction with and for which .
Proof. Let be a positive integer. Then by the pigeonhole principle, among the points (where denotes the fractional part of ), there are at least two numbers such that . Then for some integer . The statement is proved if we put .

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Reference
[1] D. Benko, "A New Approach to Hilbert's Third Problem," American Mathematical Monthly114(8), 2007 pp. 665–676.
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