Dirichlet's approximation theorem

The Dirichlet approximation theorem, named after Peter Gustav Lejeune Dirichlet, is a mathematical theorem about the quality of the approximation (approximation ) of real numbers by rational numbers.

The sentence reads: exist for each and every one and one, so that

This theorem can be proved using the drawer principle.

From the theorem follows by and in compliance with Division after that there are infinitely many pairs of positive integers, for every real, the

. meet For rational numbers almost all such approximations have the form, infinity interesting statement is therefore only for irrational numbers. The set of Hurwitz improves the inequality by a factor.

Example: Let and. Then after the Dirichlet approximation theorem (at least) one of the numbers at most of a whole number is removed. is actually