Prove The Limit Does Not Exist

Question

So I have a few questions in which I have to prove that the given sequence does not have a limit and I'm not too sure if I'm on the right track and if I am what is the next step that I have to do. Can anybody give me any feedback on my work?

Answer 1

So for your first problem--the idea of a limit--when it exists-- is that all subsequences of the original sequence will converge to it.

So, simply put, if $\lim_{n\to\infty} a_n = L$ then for all $b_{n_i}$ where $b_{n_i}$ is a subsequence defined by an indexing set $I$ with $i\in I$, $$ \lim_{i\to \infty} b_{n_i} = L $$

Thus, what you're trying to say in $(1)$ is that there's two subsequences, namely, $b_n=(-1)^{2n}$, and $c_n=(-1)^{2n+1}$, that have different limits. The contradiction proof goes like this: Assume that $a_n=(-1)^n$ has a limit-- then $b_n$ and $c_n$ converge to the same thing. But $b_n$ and $c_n$ don't converge to the same thing-- contradiction.