Prove that $\{a_n\}$ converges if $|a_n|\le 2$ and $|a_{n+2}-a_{n+1}|\le \frac{1}{8} |a_{n+1}^2-a_n^2|$

Question

$\{a_n\}$ is a sequence such that $|a_n|\le 2$ and $$|a_{n+2}-a_{n+1}|\le \frac{1}{8} |a_{n+1}^2-a_n^2|$$ How to prove that $\{a_n\}$ converges using only inequalities, Cauchy-sequences and the squeeze theorem?

I have no clue how to start with the proof. I have tried some nonsensical algebra but I think I'm stuck. Could it help if I knew what is the limit of the sequence? I would like some direction.

Answer 1

Hint:

\begin{align} |a_{n+2}-a_{n+1}|&\leq \frac{1}{8} |a_{n+1}^2-a_n^2| \\ &=\frac{1}{8}|a_{n+1}+a_n| |a_{n+1}-a_n| \\ &\leq \frac 12 |a_{n+1}-a_n| \end{align}

Hence, $|a_{n+k}-a_n|\leq ...$?