Let $\displaystyle a_n$ be a sequence of real numbers such that $\displaystyle |a_{n+1}-a_n|<\frac{1}{2^n}$ for $\displaystyle n=1,2,\dots$ prove that the sequence converges.

My first thought was to show that $\displaystyle a_n$ was a Cauchy sequence but am having trouble doing so. Any ideas about what else I should try?