Dirichlet's Approximation Theorem

Given any real number  and any positive integer , there exist integers  and  with  such that

A slightly weaker form of the theorem states that for every real , there exist integers  and  with  and  such that

REFERENCES:

Apostol, T. M. "Dirichlet's Approximation Theorem." §7.2 in Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 143-145, 1997.