# limit superior

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 $lim\; supS=\overline{lim}$, pronounced the
limit superior of $S$, to be the supremum^{} of all the limit
points of $S$. If there are no limit points, we define the limit
superior to be $-\mathrm{\infty}$.

We can generalize the above definition to the case of a mapping $f:X\to \mathbb{R}$. Now, we define a limit point of $f$ to be an $x\in \mathbb{R}$ such that for all $\u03f5>0$ there exist infinitely many $y\in X$ such that

$$ |

We then define $lim\; supf$, to be the supremum of all the limit points of $f$, or $-\mathrm{\infty}$ if there are no limit points. We recover the previous definition as a special case by considering the limit superior of the inclusion mapping $\iota :S\to \mathbb{R}$.

Since a sequence of real numbers ${x}_{0},{x}_{1},{x}_{2},,\mathrm{\dots}$ is just a
mapping from $\mathbb{N}$ to $\mathbb{R}$, we may adapt the above definition
to arrive at the notion of the limit superior of a sequence. However
for the case of sequences, an alternative, but equivalent^{} definition
is available. For each $k\in \mathbb{N}$, let ${y}_{k}$ be the supremum of
the ${k}^{\text{th}}$ tail,

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

This construction produces a non-increasing sequence

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

which either converges to its infimum^{}, or diverges to $-\mathrm{\infty}$.
We define the limit superior of the original sequence to be this limit;

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

Title | limit superior |
---|---|

Canonical name | LimitSuperior |

Date of creation | 2013-03-22 12:21:58 |

Last modified on | 2013-03-22 12:21:58 |

Owner | rmilson (146) |

Last modified by | rmilson (146) |

Numerical id | 12 |

Author | rmilson (146) |

Entry type | Definition |

Classification | msc 26A03 |

Synonym | limsup |

Synonym | supremum limit |

Related topic | LimitInferior |