Let ${a_n}$ be Cauchy sequence of real numbers. Then, there exists $alpha in (0, 1)$ such that $|a_{n+1} ā a_n| < alpha|a_n ā a_{nā1}| forall n ā„ 2$.

I want to prove or disprove this statement. I know that latter is the definition of Contractive series.

However I’m not able to show it. Neither can I think of any contradictory example… Please give me a small hint…