The hyperreal number $\{\frac{1}{n}\}_{n \in \mathbb{N}}\; \in {}^*\mathbb{R}\;$ is infinitesimal.

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

$\{n \in \mathbb{N} : 0 < \frac{1}{n}\} = \mathbb{N} \in \mathcal{F}\;\;\;\;$ so $\;\;0 < \{\frac{1}{n}\}_{n \in \mathbb{N}}$

Given any positive $a \in \mathbb{R}$ we have that $\{n \in \mathbb{N} : a \leq \frac{1}{n}\}$ is finite, so $\{n \in \mathbb{N} : \frac{1}{n} < a\} \in \mathcal{F}$ and therefore $\{\frac{1}{n}\}_{n \in \mathbb{N}} < \{a\}_{n \in \mathbb{N}}$

Thus $0 < \{\frac{1}{n}\}_{n \in \mathbb{N}} < \{a\}_{n \in \mathbb{N}}$ for every positive real number $\{a\}_{n \in \mathbb{N}} \in \mathbb{R}$ and so $\{\frac{1}{n}\}_{n \in \mathbb{N}}\;$ is infinitesimal.$\square$