TITLE: Why doesn't $d(x_n,x_{n+1})\rightarrow 0$ as $n\rightarrow\infty$ imply ${x_n}$ is Cauchy?
QUESTION [24 upvotes]: What is an example of a sequence in $\mathbb R$ with this property that is not Cauchy?
I know that Cauchy condition means that for each $\varepsilon>0$ there exists $N$ such that $d(x_p,x_q)<\varepsilon$ whenever $p,q>N$.

REPLY [2 votes]: Another example in the line with the usual metric is $x_n = \sqrt{n}$.