TITLE: Limsup and liminf of $(-1)^{n+1}+\frac{(-1)^n}{n}$. Prove your assertion.
QUESTION [0 upvotes]: I’m comfortable finding the values for limsup and liminf of the following sequence. 
$$(-1)^{n+1}+\frac{(-1)^n}{n}$$
I’m then asked to prove it. How would I go about rigorously proving $\liminf=-1$ or $\limsup=1$
My intuition tells me to use an $\varepsilon$-$N$ argument but I don’t really know where to start as my experience with those is with limits in particular. 
I know that limsup, for example, is just the limit as $n$ approaches infinity of the supremum of the tail end of our sequence, but am not sure how to proceed with a rigorous proof

REPLY [0 votes]: Might be easier to notice the sequence $$(-1)^n(\frac{1}{n}-1) = (-1)^{n+1} +\frac{(-1)^n}{n}.$$
And look at the odd indexed subsequence and the even indexed subsequence.
{"set_name": "stack_exchange", "score": 0, "question_id": 3434806}