1. ## Unbounded sets

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.

2. Originally Posted by BelaTalbot
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.
You have got tell us more than that.
Are you talking about sets of real numbers?
Are you considering $\displaystyle \pm\infty$?
Or are we in some other space altogether?

3. 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.

4. Sorry, that wasn't very clear. In the complex plane, can there be a set NOT of the format ($\displaystyle -\infty$, a] or (a, $\displaystyle \infty$) that is unbounded?

5. Originally Posted by BelaTalbot
Sorry, that wasn't very clear. In the complex plane, can there be a set NOT of the format ($\displaystyle -\infty$, a] or (a, $\displaystyle \infty$) that is unbounded?
Sure. $\displaystyle \mathbb{C}$

6. Or $\displaystyle \{ x+ iy | x= y\}$

7. 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.