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.$\mathrm{\square }$

Title	example of infinitesimal hyperreal number
Canonical name	ExampleOfInfinitesimalHyperrealNumber
Date of creation	2013-03-22 17:25:57
Last modified on	2013-03-22 17:25:57
Owner	asteroid (17536)
Last modified by	asteroid (17536)
Numerical id	4
Author	asteroid (17536)
Entry type	Example
Classification	msc 26E35