ILL 1982 Problem 22

Let $M$ be the set of real numbers of the form $\frac{m+n}{\sqrt{m^2+n^2}}$ , where $m$ and $n$ are positive integers. Prove that for every pair $x \in M, y \in M$ with $x < y$ , there exists an element $z \in M$ such that $x < z < y.$