Does the following sequence converge, if so find the limit, if not prove that no limit exists: $a_n=(-1)^nn^2$ $$a_n=(-1)^nn^2$$
I know that this sequence does not converge and hence does not have a limit. I have tried proving that this sequence does not have a limit by contradiction. 
I assumed that the limit 'a' existed and then performed the reverse triangle inequality. Thus:
$$|(-1)^nn^2-a|<\epsilon$$
$$=|(-1)^nn^2|-|a|<\epsilon$$
From here I am stuck, I have been told that I should be considering two cases for the even and odd I believe, but am not too sure how to go forward and make these contradictions.

There are different ways to see that it does not converge. One of the simplest ways is to note that your sequence $a_n$ is unbounded: It tends to $\infty$ along all even numbers $2n$, and tends to $-\infty$ along all odd numbers $2n+1$. Since a convergent sequence is bounded, this gives the result.
