example of infinitesimal hyperreal number

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

Proof - Let $ℱ$ be the nonprincipal ultrafilter in the entry (http://planetmath.org/Hyperreal).

so .

Given any positive $a\in ℝ$ we have that $\{n\in \mathbb{N}:a\le \frac{1}{n}\}$ is finite,so and therefore .

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