limit superior

Let $S\subset ℝ$ be a set of real numbers. Recall that a limitpoint 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 thelimit superior of $S$, to be the supremum of all the limitpoints of $S$. If there are no limit points, we define the limitsuperior to be $-\mathrm{\infty }$.

We can generalize the above definition to the case of amapping $f:X\to ℝ$. 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 thesupremum of all the limit points of $f$, or $-\mathrm{\infty }$ if there are nolimit points. We recover the previous definition as a special case byconsidering 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 amapping from $\mathbb{N}$ to $ℝ$, we may adapt the above definitionto arrive at the notion of the limit superior of a sequence. Howeverfor the case of sequences, an alternative, but equivalent definitionis available. For each $k\in \mathbb{N}$, let $y_k$ be the supremum ofthe $k^{\text{th}}$ tail,

$$y_k=\underset{j\ge k}{sup}{x}_j.$$

This construction produces anon-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;

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