$S=\{5+\frac{(-1)^n}{n} \; : \; n \in \mathbb N\}$
According to my calculations this set has a lower bound of $4$ and an upperbound of $5$; however, since $4$ is reachable by the set it is a minimum and an infimum. Since $5$ is the limit of the sequence it is not reachable and there is no maximum in the set, yet the supremum is $5$.
So I say the set is neither open nor closed and is the interval $[4,5)$
Is this correct?
It is not closed since there is a sequence of elements in this set which converge to $5$, yet $5$ is not in this set.
Notice that it is also not open set. If $S \subset \mathbb{R}$ is open, then for every point in $x\in S$, there exists a ball $B_{\epsilon}(x)$ such that $B_\epsilon(x)\subset S$. This is clearly not true by taking $x = 4$.