# limit inferior

Let $S\subset \mathbb{R}$ be a set of real numbers. Recall that a limit
point^{} of $S$ is a real number $x\in \mathbb{R}$ such that for all
$\u03f5>0$ there exist infinitely many $y\in S$ such that

$$ |

We define $lim\; infS$, pronounced the
limit inferior of $S$, to be the infimum^{} of all the limit
points of $S$. If there are no limit points, we define the limit
inferior to be $+\mathrm{\infty}$.

The two most common notations for the limit inferior are

$$lim\; infS$$ |

and

$$\underset{\xaf}{lim}S.$$ |

An alternative, but equivalent^{}, definition is available in the case of
an infinite^{} sequence^{} of real numbers ${x}_{0},{x}_{1},{x}_{2},,\mathrm{\dots}$. For
each $k\in \mathbb{N}$, let ${y}_{k}$ be the infimum of the ${k}^{\text{th}}$ tail,

$${y}_{k}=\underset{j\ge k}{inf}{x}_{j}.$$ |

This construction produces a non-decreasing sequence

$${y}_{0}\le {y}_{1}\le {y}_{2}\le \mathrm{\dots},$$ |

which either converges to its supremum, or diverges to $+\mathrm{\infty}$. We define the limit inferior of the original sequence to be this limit;

$$\underset{k}{lim\; inf}{x}_{k}=\underset{k}{lim}{y}_{k}.$$ |

Title | limit inferior |
---|---|

Canonical name | LimitInferior |

Date of creation | 2013-03-22 12:22:01 |

Last modified on | 2013-03-22 12:22:01 |

Owner | rmilson (146) |

Last modified by | rmilson (146) |

Numerical id | 10 |

Author | rmilson (146) |

Entry type | Definition |

Classification | msc 26A03 |

Synonym | liminf |

Synonym | infimum limit |

Related topic | LimitSuperior |