# 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).

$$ so $$.

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

Thus $$ for every positive real number ${\{a\}}_{n\in \mathbb{N}}\in \mathbb{R}$, 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 |