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 $$\left |\xi-\frac{m}{n}\right | < \frac{1}{\sqrt{5}\, n^2}.$$

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

The theorem is equivalent to the claim that the Markov constant of every number is larger than $$\sqrt{5}$$.