Approximation of Irrationals
If is irrational and is any positive integer, there is a fraction with and for which .[more]
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 .[less]
 D. Benko, "A New Approach to Hilbert's Third Problem," American Mathematical Monthly, 114(8), 2007 pp. 665–676.