PlanetMath: limit superior

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limit superior

(Definition)

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

$\displaystyle \vert x-y\vert <\epsilon.$

We define $\limsup S=\overline{\lim}$ , pronounced the limit superior of

, to be the supremum of all the limit points of

. If there are no limit points, we define the limit superior to be $-\infty$ .

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

$\displaystyle \vert x-f(y)\vert <\epsilon.$

We then define $\limsup f$ , to be the supremum of all the limit points of

, or $-\infty$ 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 $x_0, x_1, x_2, ,\ldots$ 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 be the supremum of the $k^{\text{th}}$ tail,

$\displaystyle y_k = \sup_{j\geq k} x_j .$

This construction produces a non-increasing sequence

$\displaystyle y_0 \geq y_1 \geq y_2 \geq \ldots,$

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

$\displaystyle \limsup_{k} x_k = \lim_k y_k.$

"limit superior" is owned by rmilson.

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