The hyperreal number $\{n\}_{n \in \mathbb{N}}\; \in {}^*\mathbb{R}\;$ is infinite (or unlimited).

Proof : Let $\mathcal{F}$ be the nonprincipal ultrafilter fixed in the parent entry.

Given any positive $a \in \mathbb{R}$ we have that $\{n \in \mathbb{N} : n \leq a\}$ is finite, so $\{n \in \mathbb{N} : a < n\} \in \mathcal{F}$ and therefore $\{a\}_{n \in \mathbb{N}} < \{n\}_{n \in \mathbb{N}}$

Thus $\{n\}_{n \in \mathbb{N}}$ is infinite.$\square$