Discretized Rotation with fixed initial points

We prove that if $\max \{\mathrm{\mid }{a}_0\mathrm{\mid },\mathrm{\mid }{a}_1\mathrm{\mid }\}\le 10$ and $\lambda \in ]-2,2[$, then the sequence defined by $$0\le a_{n+2}+\lambda a_{n+1}+a_n<1$$ is periodic.