SCHWEITZER 2005 Problem 2

Let $(a_{n})_{n \ge 1}$ be a sequence of integers satisfying the inequality $$0\le a_{n-1}+\frac{1-\sqrt{5}}{2}a_{n}+a_{n+1} <1$$ for all $n \ge 2$ . Prove that the sequence $(a_{n})$ is periodic.

Any Hints or Sols for this hard problem?? :help: