# convergence of a sequence with finite upcrossings

The following result characterizes convergence of a sequence in terms of finiteness of numbers of upcrossings.

###### Theorem.

A sequence ${x}_{\mathrm{1}}\mathrm{,}{x}_{\mathrm{2}}\mathrm{,}\mathrm{\dots}$ of real numbers converges to a limit in the extended real numbers if and only if the number of upcrossings $U\mathit{}\mathrm{[}a\mathrm{,}b\mathrm{]}$ is finite for all $$.

Since the number of upcrossings $U[a,b]$ differs from the number of downcrossings $D[a,b]$ by at most one, the theorem can equivalently be stated in terms of the finiteness of $D[a,b]$.

Title | convergence of a sequence with finite upcrossings |
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Canonical name | ConvergenceOfASequenceWithFiniteUpcrossings |

Date of creation | 2013-03-22 18:49:36 |

Last modified on | 2013-03-22 18:49:36 |

Owner | gel (22282) |

Last modified by | gel (22282) |

Numerical id | 4 |

Author | gel (22282) |

Entry type | Theorem |

Classification | msc 40A05 |

Classification | msc 60G17 |

Related topic | UpcrossingsAndDowncrossings |