Prove that the sequence :
$\displaystyle x_n = n^2 + \frac{(-1)^n}{n}$ is not cauchy
If it were Cauchy then for any $\displaystyle \epsilon >0\,\,\,\exists M\in\mathbb{N}\,\,\,s.t.\,\,\,|x_n-x_m|<\epsilon\,\,\,\forall\,n,m>M$ . In particular, this must be true for $\displaystyle m=n+1$ , but:
$\displaystyle |x_n-x_{n+1}|=\left|-2n-1+\frac{(-1)^n}{n(n+1)}\right|=2n+1\pm \frac{1}{n(n+1)}>2$ , so it is enough to take $\displaystyle \epsilon < 2$ and the above definitory property of Cauchy sequences won't be true for it.
Tonio