If $\{x_n\}$ is a Cauchy sequence, does that imply $\sum_{i=1}^{\infty}d(x_i,x_{i+1})< \infty ?$

Question

If $\{x_n\}$ is a Cauchy sequence, does that imply $\sum_{i=1}^{\infty}d(x_i,x_{i+1})< \infty ?$

I feel like answer should be NO but I am unable to find such an exapmle. Can anybody please help me?

Answer 1

Hint: Take any conditionally convergent but not absolutely convergent series $\sum_{n\geq 1}a_n$ and define $$ x_N = \sum_{n=1}^{N}a_n.$$ $a_n=\frac{(-1)^n}{n}$ does the job pretty fine.