TR-MO 1993 Problem 4

$a_{n}$ is a sequence of positive integers such that, for every $n\geq 1$ , $0<a_{n+1}-a_{n}<\sqrt{a_{n}}$ . Prove that for every $x,y\in{R}$ such that $0<x<y<1$ $x< \frac{a_{k}}{a_{m}}<y$ we can find such $k,m\in{Z^{+}}$ .