I have the sequence $$(a_n)_{n \in {\mathbb{N}}} = \lim_{n\to\infty} \frac{\frac{n!}{n^n}+1}{\frac{n!}{n^n}+(-1)^n}$$ I can see intuitively why this doesn't converge as it acts like $(-1)^n$ for large $n$, which doesn't converge, but I'm unsure how to show formally that it doesn't converge?
I think assuming it has a limit, say $L$, and maybe choosing a certain $\epsilon$ so that $|a_n - L| < \epsilon$, then trying a few different cases of $a_n$'s to try and find a contradiction regarding the limit is the way to go, but I have no idea what $\epsilon$ to choose or how to construct the proof.
The easiest way is to consider the subsequences $(a_{2n})$ and $(a_{2n+1})$, and show that the first one converges to $1$ and the second one to $-1$.