Unbounded sets

**BelaTalbot** · March 5th 2010, 06:28 AM

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.

**Plato** · March 5th 2010, 06:48 AM

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 $\pm\infty$ ?
Or are we in some other space altogether?

**Pinkk** · March 5th 2010, 07:39 AM

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 $+\infty$ or $-\infty$ . Then the set $S = \{(-1)^{n}n : n \in \mathbb{N}\}$ is clearly unbounded but $\lim_{n\to \infty} (-1)^{n}n$ does not exist.

**BelaTalbot** · March 5th 2010, 08:00 AM

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

**Drexel28** · March 5th 2010, 08:00 AM

Originally Posted by BelaTalbot

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

Sure. $\mathbb{C}$

**HallsofIvy** · March 5th 2010, 08:13 AM

Or $\{ x+ iy | x= y\}$

**Plato** · March 5th 2010, 08:24 AM

Suppose that $\mathcal{M}\subseteq\mathbb{C}$ having the property that for each $N\in \mathbb{Z}^+$ there is some $z\in \mathcal{M}$ such that $|z|\ge N$ .
In that case the set $\mathcal{M}$ is unbounded.

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