I am trying to show that if $A \subseteq \mathbb{R}$ is closed, then, for a sequence $x_n \in A$ is such that $\sum_{n \in \mathbb{N}} d(x_n,x_{n+1}) < \infty$, then $x_n$ converges in A.
Essentially I must use the series convergence to show that there is a limit of the sequence, and want to use cauchy criterion. By that convergence, $\sum_{i = n}^{n+k} \vert \vert x_i - x_{i+1} \vert \vert \leq \vert \vert x_n \vert \vert - \vert \vert x_{n+1} \vert \vert + \vert \vert x_{n+1} \vert \vert - \vert \vert x_{n-2} \vert \vert + \vert \vert x_{n+2} \vert \vert \dots - \vert \vert x_{n+k}\vert \vert =$
$ \vert \vert x_n \vert \vert - \vert \vert x_{n+k} \vert \vert < \epsilon$ and by completeness property $x_n \rightarrow x \in \mathbb{R}$, and since $A$ is closed, $x \in A$. Is there anything I'm missing or I have gotten wrong? It seems as if I may have skipped something in showing $x_n$ converges.
Hint: You want to show the sequence $x_n$ is Cauchy. Observe we have \begin{align} d(x_n, x_{n+m}) \leq \sum^{n+m}_{i=n} d(x_i, x_{i+1}). \end{align} I will let you finish the rest.