Using definition prove that $\frac{1}{n}+(-1)^n$ is not cauchy 
Using definition prove that $\frac{1}{n}+(-1)^n$ is not cauchy

Let $n,m \in \mathbb{N}$ and $m=2n+1$ then
$|x_{m}-x_{2n}|=|x_{2n+1}-x_{2n}|=\bigg|\frac{1}{2n+1}-1-\frac{1}{2n}-1\bigg|=\bigg|2+\frac{1}{2n}-\frac{1}{2n+1}\bigg|>2$
thus by definition contradic the definition given an $\epsilon > 0$ we have a $N \in \mathbb N$ such that for all $n,m > N$ we have that $|x_n - x_m| < \epsilon$
Can anyone verify my answer?

Your answer is right since, as you say, if $\varepsilon\leq 2$ the definition is not fulfilled.
