Simultaneous Approximation of Two Real Numbers by Rationals

This Demonstration illustrates the simultaneous approximation of two real numbers by rational numbers, because there is always a small square with at least two points.



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Lemma 1. Let be real numbers. For each it is possible to approximate simultaneously by rational numbers , in the sense that (). In addition, if all are positive, then the can be chosen to be positive, where is replaced with .
Proof. Let be a positive integer satisfying . Then by the pigeonhole principle, among the points (), where denotes the fractional part of , there are at least two numbers such that for all . Then for some integers . The statement is proved if we put .
This lemma was used in the elementary proof of Hilbert's third problem.
[1] D. Benko, "A New Approach to Hilbert's Third Problem," American Mathematical Monthly  114(8), 2007 pp. 665–676.
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