Let $x_n = \sqrt n $, $\forall p \in \mathbb{N} ,\forall ε>0,\exists N=[\frac{p^2}{ε^2}]+1,n>N$, $|x_{n+p}-x_n|=\frac{p}{\sqrt{n+p}+\sqrt n}<\frac{p}{\sqrt n}<ε$
So the sequence$\{x_n\}$ is a Cauchy sequence? Then the sequence is convergent? But the sequence $\{x_n\}$, in fact, is divergent.