AUSTRIAN-POLISH 1981 Problem 6

The sequences $(x_n), (y_n), (z_n)$ are given by $x_{n+1}=y_n +\frac{1}{x_n}$ , $y_{n+1}=z_n +\frac{1}{y_n}$ , $z_{n+1}=x_n +\frac{1}{z_n}$ for $n \ge 0$ where $x_0,y_0, z_0$ are given positive numbers. Prove that these sequences are unbounded.