If ${a_n}$ is a Cauchy sequence of real numbers. Then, is the following statement always true?

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

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…