ILL 1976 Problem 20

Let $(a_n), n = 0, 1, . . .,$ be a sequence of real numbers such that $a_0 = 0$ and $$a^3_{n+1} = \frac{1}{2} a^2_n -1, n= 0, 1,\cdots$$ Prove that there exists a positive number $q, q < 1$ , such that for all $n = 1, 2, \ldots ,$ $$|a_{n+1} - a_n| \leq q|a_n - a_{n-1}|,$$ and give one such $q$ explicitly.