# Do all closed, unbounded set have something to do with infinity? [closed]

By ‘have something to do with infinity’ I mean things like the limit of its subsequence goes to infinity. If not is there any counter example?

• What ambient space are your sets taken from? If they are subsets of $\Bbb R$, for example, then it is true that a set (closed or not) is unbounded if and only if it contains a sequence $\{x_n\}$ such that $\lim_{n\to\infty} |x_n| = \infty$. May 14 at 0:42
• Yes I’m assuming real number. Thanks for the clarification. May 14 at 2:25