# example of infinite hyperreal number

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

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

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 and therefore $\{a\}_{n\in\mathbb{N}}<\{n\}_{n\in\mathbb{N}}$.

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

Title example of infinite hyperreal number ExampleOfInfiniteHyperrealNumber 2013-03-22 17:25:59 2013-03-22 17:25:59 asteroid (17536) asteroid (17536) 5 asteroid (17536) Example msc 26E35