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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

Reference

[1] D. Benko, "A New Approach to Hilbert's Third Problem," American Mathematical Monthly, 114(8), 2007 pp. 665–676.

Contributed by: Izidor Hafner

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Approximation of Irrationals