If a sequence $(x\_n)\_{n=1}^{\infty}$ in $\mathbb{R}^n$ satisfies $\sum\_{n\geq 1} ||x\_n-x\_{n+1}||<\infty$, show that it is Cauchy.

This isn't a complete answer, but here's my train of thought. Since $\sum\_{n\geq 1} ||x\_n-x\_{n+1}||$ is finite, $\lim\_{n\to\infty}||x\_n-x\_{n+1}|| = 0$. This implies that $(x\_n)$ is a monotone decreasing sequence. Next, we can find an upper bound for $x\_n$. After showing that $x\_n$ is bounded, we know it is a bounded monotone sequence so it converges and is therefore a Cauchy sequence.

I know I would still need to prove that there is an upper bound, but does this generally make sense of have I missed something? Thanks!