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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?

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    Let $x_n$ be the partial sums of the [edit: alternating] harmonic series. – 2017-02-09
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    LeBtz : who would have thought that the harmonic series was a Cauchy sequence.. – 2017-02-09
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    Eh i mean the alternating one. Sorry – 2017-02-09

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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.

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    THE canonical example (+1) – 2017-02-09