Consider $\{(−1)^n + 1/n : n \in \Bbb N\setminus \{0\}\}\subset \Bbb R$. Is this set open or closed?

Question

Consider $\{(−1)^n + 1/n : n \in \Bbb N\setminus \{0\}\}\subset \Bbb R$. Is this set open or closed?

As this is a union of discrete points, it should be closed but in the set limit of convergent sequence $1+\frac{1}{2n}$, which is $1$, does not belong to this set, so is this set open, closed or neither of the two?

Answer 1

When $n=1$ the term is $0,$ yet the set doesn't contain any open interval around $0.$ So the set is not open. You already showed it isn't closed.