limit superior

Let $S\subset ℝ$ be a set of real numbers. Recall that a limit point of $S$ is a real number $x\in ℝ$ such that for all $ϵ>0$ 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 $f:X\to ℝ$. Now, we define a limit point of $f$ to be an $x\in ℝ$ such that for all $ϵ>0$ 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 $\iota :S\to ℝ$.

Since a sequence of real numbers $x_0,x_1,x_2,,\mathrm{\dots }$ is just a mapping from $\mathbb{N}$ to $ℝ$, 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