Hurwitz's theorem (number theory)

In number theory, Hurwitz's theorem, named after Adolf Hurwitz, gives a bound on a Diophantine approximation. The theorem states that for every irrational number ξ there are infinitely many relatively prime integers m, n such that

The hypothesis that ξ is irrational cannot be omitted. Moreover the constant 5 is the best possible; if we replace 5 by any number A > 5 and we let ξ = ( 1 + 5 ) / 2 (the golden ratio) then there exist only finitely many relatively prime integers m, n such that the formula above holds.