Approximations to two real numbers

For a real $\xi $put $\psi_\xi (t) = \min_{1 \le x \le t}|| x \xi ||$. Let $\alpha , \beta$ be real numbers such that $\alpha \pm \beta \not \in \mathbb{Z}$. We prove that the function $\psi_\alpha (t)-\psi_\beta (t)$ changes its sign infinitely many often as $t \to + \infty$. The proof uses continued fractions.