TITLE: Cauchy sequence and sums of distances of consecutive terms
QUESTION [5 upvotes]: Consider a sequence $(x_n)$ in metric space $(X,d)$. 
If the sequence of partial sums $\sum_{i=1}^n d(x_i,x_{i+1})$ of distances between consecutive terms converges in $\mathbb{R}$, then I understand why the sequence is Cauchy. 
But is the converse statement true as well?

REPLY [7 votes]: No. A counterexample is afforded by the sequence $a_n=(-1)^n/n$ in $\mathbb R$ with the standard metric, which is Cauchy but for which the sum of distances is essentially twice the harmonic series and thus diverges.