This proof is in my notes. I would appreciate any guidance that anyone has to offer.

Let $\displaystyle (x_n)$ be a sequence of complex numbers. If there is a constant $\displaystyle c<1$ such that $\displaystyle |x_{n+1}-x_n|\leq c|x_n-x_{n-1}|$ for all $\displaystyle n\geq 2$, prove that $\displaystyle (x_n)$ is convergent.

Judging by where this is in my notes I'm assuming I might be supposed to use the definition of a Cauchy sequence to prove this.