Let {${x\_n}$} be a sequence such that there exists a $0<C<1$, such that:
$|x\_{n+1}-x\_n|\leq$$C|x\_n-x\_{n-1}|$. Prove that {$x\_n$} is a Cauchy Sequence. 

I'm given a hint that $1+C+C^2...+C^n=\frac{1-C^{n+1}}{1-C}$, which I've used to construct:

$|x\_m-x\_n|\leq |x\_n-x\_{n-1}|\frac{1-C^{k+1}}{1-C}$ 

where $k=m-n$,but I'm not sure how to use the definition of a Cauchy sequence to proceed. Any help is greatly appreciated.