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I am trying to solve following problem from Apostol's Mathematical Analysis. The problem could be very trivial, but I am not getting clue for it.

Let $\{a_n\}$ be a sequence of real numbers in $[-2,2]$ such that $$|a_{n+2}-a_{n+1}|\le \frac{1}{8} |a_{n+1}^2-a_n^2| \,\,\,\, \mbox{ for all } n\ge 1.$$ Prove that $\{a_n\}$ is convergent.

Q. Any hint for solving this? (I was not getting the restrictions of interval and the factor $\frac{1}{8}$).

My try: since $a_i\in [-2,2]$ so $a_i^2\in [0,4]$. Thus, $|a_{n+1}^2-a_n^2|\le 4$ and so $|a_{n+2}-a_{n+1}|\le \frac{1}{2}$. After this, I couldn't proceed.

Any HINT is sufficient.