IMC 2018 Problem 7

Let $(a_n)_{n=0}^{\infty}$ be a sequence of real numbers such that $a_0=0$ and $$a_{n+1}^3=a_n^2-8 \text{for} n=0,1,2,…$$ Prove that the following series is convergent: $$\sum_{n=0}^{\infty}{|a_{n+1}-a_n|}.$$ Proposed by Orif Ibrogimov, National University of Uzbekistan