# sequence of sets convergence

Let ${\left\{{A}_{n}\right\}}_{n=1}^{\mathrm{\infty}}$ be a sequence of sets, and $A$ a set.

The sequence ${\left\{{A}_{n}\right\}}_{n=1}^{\mathrm{\infty}}$ is said to * from below* to $A$, (shortly, ${A}_{n}\uparrow A$ or ${A}_{n}\nearrow A$), iff

1) ${A}_{n}\subseteq {A}_{n+1}$ $\forall n\ge 1$

2) $A={\displaystyle \bigcup _{n=1}^{\mathrm{\infty}}}{A}_{n}$

The sequence ${\left\{{A}_{n}\right\}}_{n=1}^{\mathrm{\infty}}$ is said to * from above* to $A$, (shortly, ${A}_{n}\downarrow A$ or ${A}_{n}\searrow A$), iff

1) ${A}_{n+1}\subseteq {A}_{n}$ $\forall n\ge 1$

2) $A={\displaystyle \bigcap _{n=1}^{\mathrm{\infty}}}{A}_{n}$

In both cases the less accurate notation

$$A=\underset{n\u27f6\mathrm{\infty}}{lim}{A}_{n}$$ |

is also used.

Title | sequence of sets convergence |
---|---|

Canonical name | SequenceOfSetsConvergence |

Date of creation | 2013-03-22 16:15:12 |

Last modified on | 2013-03-22 16:15:12 |

Owner | Andrea Ambrosio (7332) |

Last modified by | Andrea Ambrosio (7332) |

Numerical id | 8 |

Author | Andrea Ambrosio (7332) |

Entry type | Definition |

Classification | msc 28A05 |