I came across another question about infimums of sets.
Find $\inf \{(-1)^{n} + 1/n: n \in \mathbb{Z}^{+} \}$.
Heuristaically, I think the infimum is $-1$ since, for large $n$, the second term goes to $0$ and the first term is at least $-1$. Let $A = \{(-1)^{n} + 1/n: n \in \mathbb{Z}^{+} \}$. I know that $-1$ is a lower bound for $A$ (would I have to justify this?). So $\alpha = \inf A$ exists. Thus $-1 \leq \alpha$. Now I need to show that $\alpha \leq -1$. Suppose for contradiction that $\alpha > -1$. Then I would need to find a $\beta$ such that $\beta > \alpha > -1$ where $\beta$ is also a lower bound?