A sequence of real numbers such that $\lim_{n\to+\infty}|x_n-x_{n+1}|=0$ but it is not Cauchy Give an example of a sequence $(x_n)$ of real numbers, where
$\displaystyle\lim_{n\to+\infty}|x_n-x_{n+1}|=0$, but $(x_n)$ is not a Cauchy sequence

Take $x_n=\sqrt n$. Since $|x_{n+1}-x_n|=\left|\sqrt{n+1}-\sqrt n\right|=\left|\frac {n+1-1}{\sqrt{n+1}+\sqrt n}\right|\leq \frac 1{\sqrt n}$, we have $\displaystyle\lim_{n\to+\infty}x_{n+1}-x_n=0$. But we have $|x_{2n}-x_n|=\sqrt  2\sqrt n-\sqrt n=(\sqrt 2-1)\sqrt n$, hence $(x_n)_n$ cannot be a Cauchy sequence.
