Let $a_n$ be a sequence s.t: $|a_n| \lt 2$ and $|a_{n+2}-a_{n+1}| \lt \frac{|a^2_{n+1}-a^2_{n}|}{5}$
Prove that $a_n$ converges.
Well, by intuintion i can feel that this screams "cauchy sequence", meaning that from some index N and for every m,n>N : $|a_m-a_n|\lt \epsilon$
All that being said, i don't have any clue on where or how to start this. Any hint or guidance to the right direction would be great. Thanks!
$|a_{n+2}-a_{n+1}| <r |a_{n+1}-a_{n}|$ where $r=\frac 4 5$ (because $a_{n+1}^{2}-a_n^{2}=(a_{n+1}-a_{n})(a_{n+1}+a_{n})$. Can you finish the proof by showing that $(a_n)$ is Cauchy?