# uniqueness of limit of sequence

If a number sequence has a limit, then the limit is uniquely determined.

*Proof.* For an indirect proof (http://planetmath.org/ReductioAdAbsurdum), suppose that a sequence

$${a}_{1},{a}_{2},{a}_{3},\mathrm{\dots}$$ |

has two distinct limits $a$ and $b$. Thus we must have both

$$ |

and

$$ |

But when $n$ exceeds the greater of ${n}_{1}$ and ${n}_{2}$, we can write

$$ |

This inequality^{} an impossibility, whence the antithesis made in the begin is wrong and the assertion is .

Title | uniqueness of limit of sequence |
---|---|

Canonical name | UniquenessOfLimitOfSequence |

Date of creation | 2013-03-22 19:00:23 |

Last modified on | 2013-03-22 19:00:23 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 4 |

Author | pahio (2872) |

Entry type | Theorem |

Classification | msc 40A05 |