According to Hurwitz Theorem (Number Theory), for every irrational number $x$ there are infinitely many relatively prime integers $m$ and $n$ such that $$\left|x - \frac{m}{n}\right| < \frac{1}{\sqrt{5}n^2}.$$
I want to show that no constant larger than $\sqrt{5}$ is the best possible, in the sense that there exists an irrational for which the inequality would not have infinitely many solutions if $\sqrt{5}$ were replaced by a larger constant.
Following Wikipedia, if we replace $\sqrt{5}$ by any number $A > \sqrt{5}$ and let $x = \frac{1+\sqrt{5}}{2}$ then there exist only finitely many relatively prime integers $m$ and $n$ such that the formula above holds. Why do there only exist finitely many in this assertion? Also, what other counterexamples come to mind?