TSTST 2012 Problem 5

A rational number $x$ is given. Prove that there exists a sequence $x_{0}, x_{1}, x_{2}, \ldots$ of rational numbers with the following properties: (a) $x_{0}=x$; (b) for every $n \geq 1$, either $x_{n}=2 x_{n-1}$ or $x_{n}=2 x_{n-1}+\frac{1}{n}$; (c) $x_{n}$ is an integer for some $n$.