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.

If you want to work directly from the definition of a convergent sequence:
Suppose that the sequence converges to $L$.  Consider $n=2k$ and $n=2k+1$.  For $k$ sufficiently large, $|a_{2k}-L|< \varepsilon$ and $|a_{2k+1}-L|<\varepsilon$.
Therefore, $|a_{2k}-a_{2k+1}|\leq |a_{2k}-L|+|a_{2k+1}-L|<2\varepsilon$.  Now, choose a $k$ so large that this is impossible.
