What is an example of an unbounded set that doesn't go to infinity? I'm having serious trouble grasping whether or not a set that doesn't go to infinity is unbounded.
Waht do you mean by "go to infinity"? If you're talking about real numbers and if the set is listed as a sequence, the limit isn't $\displaystyle +\infty$ or $\displaystyle -\infty$. Then the set $\displaystyle S = \{(-1)^{n}n : n \in \mathbb{N}\}$ is clearly unbounded but $\displaystyle \lim_{n\to \infty} (-1)^{n}n$ does not exist.
Suppose that $\displaystyle \mathcal{M}\subseteq\mathbb{C}$ having the property that for each $\displaystyle N\in \mathbb{Z}^+$ there is some $\displaystyle z\in \mathcal{M}$ such that $\displaystyle |z|\ge N$.
In that case the set $\displaystyle \mathcal{M}$ is unbounded.