Proving that $(x_n)_{n=1}^{\infty}$ is a Cauchy sequence. Let $(x_n)_{n=1}^{\infty}$ be a sequence such that $|x_{n+1} - x_n| < r^n$ for all $n \geq 1$, for some $0 < r < 1$. Prove that $(x_n)_{n=1}^{\infty}$ is a Cauchy sequence.
I understand that a Cauchy sequence means that for all $\varepsilon > 0$ $\exists N$ so that for $n,m \ge N$ we have $|a_m - a_n| \le \varepsilon$. But this one is really giving me a headache.
I tried doing something like: let $ m > n$. Therefore $x_n - x_m$ =  $(x_n - x_{n-1}) + (x_{n-1} - x_{n-2}) + ... + (x_{m+1} - x_m) $ and then somehow using the triangle inequality to compute some sum such that $x_n - x_m$  < sum which would be epsilon?
any help is appreciated, thank you.

This is exactly that:
$$
x_{n+p} - x_n = \sum_{k=1}^p x_{n_k} - x_{n_{k-1}}
$$
$$
|x_{n+p} - x_n| \le  \sum_{k=1}^p |x_{n+k} - x_{{n+k-1}}| 
\le \sum_{k=1}^p r^{n+k-1} 
\\  \sum_{k=1}^\infty r^{n+k-1} 
= \frac{r^{n}}{1-r}
\to_{n\to\infty} 0
$$
So eventually, 
$$\sup_{p\in\mathbb N} |x_{n+p} - x_n| \to_{n\to\infty} 0
$$
