# limit inferior

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 $\epsilon>0$ there exist infinitely many $y\in S$ such that

 $|x-y|<\epsilon.$

We define $\liminf S$, pronounced the limit inferior of $S$, to be the infimum of all the limit points of $S$. If there are no limit points, we define the limit inferior to be $+\infty$.

The two most common notations for the limit inferior are

 $\liminf S$

and

 $\underline{\lim}\,S\,.$

An alternative, but equivalent, definition is available in the case of an infinite sequence of real numbers $x_{0},x_{1},x_{2},,\ldots$. For each $k\in\mathbb{N}$, let $y_{k}$ be the infimum of the $k^{\text{th}}$ tail,

 $y_{k}=\inf_{j\geq k}x_{j}.$

This construction produces a non-decreasing sequence

 $y_{0}\leq y_{1}\leq y_{2}\leq\ldots,$

which either converges to its supremum, or diverges to $+\infty$. We define the limit inferior of the original sequence to be this limit;

 $\liminf_{k}x_{k}=\lim_{k}y_{k}.$
Title limit inferior LimitInferior 2013-03-22 12:22:01 2013-03-22 12:22:01 rmilson (146) rmilson (146) 10 rmilson (146) Definition msc 26A03 liminf infimum limit LimitSuperior