Let $\{ x_{n}\}$ be a sequence such that there exists $0 < C < 1$ such that $|x_{n+1} - x_{n}| \leq C|x_{n} - x_{n-1}|$. Show $\{x_{n}\}$ is Cauchy.

Question

I'm stuck, this is what I've tried so far:

$|x_{n} - x_{m}| = |x_{n} - x_{n-1} + x_{n-1} -x_{n-2} + x_{n-2} + ... + x_{m}| \leq$ $|x_{n} - x_{n-1}| + |x_{n-1} -x_{n-2}| + ... +|x_{m+1} - x_{m}| \leq (1 + C + ... + C^{n})\cdot |x_{m+1} - x_{m}| = \frac{1-C^{n+1}}{ 1 - C} |x_{m+1} - x_{m}| $

And now I'm stuck trying to show $\forall \varepsilon > 0$ $\exists M$ with $n\geq M$ that implies that it can be $< \varepsilon$.