# real analysis – Proof for the Cauchy sequence

Q

Prove the following :

$$({}^exists rin(0,1) s.t. {}^forall nin mathbb{N},|a_{n+2}-a_{n+1}|leq r|a_{n+1}-a_{n}|)Rightarrow$$The sequence $${a_n}$$ is a Cauchy sequence

I understand that $$|a_{n+2}-a_{n+1}|leq r^n|a_{2}-a_{1}|$$,but I don’t found the arbitrariness of the Cauchy sequence and how to choose a good epsilon.

Thanks,for help.