# limit of nondecreasing sequence

Theorem. A monotonically nondecreasing sequence^{} of real numbers with upper bound^{} a number $M$ converges to a limit which does not exceed $M$.

Proof. Let ${a}_{1}\leqq {a}_{2}\leqq \mathrm{\dots}\leqq {a}_{n}\leqq \mathrm{\dots}\leqq M$. Therefore the set $\{{a}_{1},{a}_{2},\mathrm{\dots}\}$ has a finite supremum $s\leqq M$. We show that

$\underset{n\to \mathrm{\infty}}{lim}{a}_{n}=s.$ | (1) |

Let $\epsilon $ an arbitrary positive number. According to the definition of supremum we have ${a}_{n}\leqq s$ for all $n$ and on the other hand, there exists a member ${a}_{n(\epsilon )}$ of the sequence that is $>s-\epsilon $. Then we have $$, and since the sequence is nondecreasing,

$$ |

Thus the equation (1) and the whole theorem has been proven.

For the nonincreasing sequences there is the corresponding

Theorem. A monotonically nonincreasing sequence of real numbers with lower bound a number $L$ converges to a limit which is not less than $L$.

Note. A good application of the latter theorem is in the proof that Euler’s constant exists.

Title | limit of nondecreasing sequence |

Canonical name | LimitOfNondecreasingSequence |

Date of creation | 2013-03-22 17:40:31 |

Last modified on | 2013-03-22 17:40:31 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 13 |

Author | pahio (2872) |

Entry type | Theorem |

Classification | msc 40-00 |

Synonym | nondecreasing sequence with upper bound |

Synonym | limit of increasing sequence |

Related topic | MonotonicallyIncreasing |

Related topic | MonotoneIncreasing |

Related topic | Supremum |

Related topic | Infimum |

Related topic | ConvergenceOfTheSequence11nn |