I have the following set:
$A = {(-1)^n + (\frac{2}{n}))}$ and I am curious on a formal way of finding both the limit points and whether or not the set is open.
My stab at it is that the limit points are -1,1 because there are two subsets in this set (-1,-1,-1....) when n is large and also (1,1,1...) when n is large. Therefore the limit points are -1 and 1. Also, the set is open because of the $(\frac{2}{n})$ term which goes on and on as $n \Rightarrow \infty$
Defined like this $A$ is no set.
I suspect you mean $A:=\{(-1)^n+\frac2{n}:n=1,2,\dots\}$.
Then $-1,1$ are indeed (the only) limitpoints of $A$.
Note that $A$ is not empty. So if it is open then must contain some open interval. This would imply that $A$ is uncountable, wich is not the case.
We conclude that $A$ is not open.