I want to show that the sequence $$a_n=\frac{1}{n}+(-1)^n$$ does not converge to a limit.
I know that if a sequence $\left( a_n\right)_{n \in \mathbb{N}}$ converges to a limit L, then $\forall\epsilon >0\;\;\exists N\in\Bbb N \;\text{such that}\;\forall n\geq N, \;\;|a_n-L|\leq\epsilon$, but I am not sure how to use this to prove a sequence that does not converge to a limit.
Please give me some suggestions on where can I get started and how to write out the proof, thanks to anyone who can help.