limit inferior

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

The two most common notations for the limit inferior are

$$lim\; infS$$

and

$$\underset{¯}{lim}S.$$

An alternative, but equivalent, definition is available in the case ofan infinite sequence of real numbers $x_0,x_1,x_2,,\mathrm{\dots }$. Foreach $k\in \mathbb{N}$, let $y_k$ be the infimum of the $k^{\text{th}}$ tail,

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

This construction produces anon-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;

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