example of infinitesimal hyperreal number

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

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

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

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