# Definition:Limit Inferior of Sequence of Sets

## Definition

### Definition 1

Let $\set {E_n : n \in \N}$ be a sequence of sets.

Then the limit inferior of the sequence, denoted $\displaystyle \liminf_{n \mathop \to \infty} \ E_n$, is defined as:

 $\displaystyle \liminf_{n \mathop \to \infty} \ E_n$ $:=$ $\displaystyle \bigcup_{n \mathop = 0}^\infty \ \bigcap_{i \mathop = n}^\infty E_n$ $\quad$ $\quad$ $\displaystyle$ $=$ $\displaystyle \paren {E_0 \cap E_1 \cap E_2 \cap \ldots} \cup \paren {E_1 \cap E_2 \cap E_3 \cap \ldots} \cup \cdots$ $\quad$ $\quad$

### Definition 2

Let $\left\{{E_n : n \in \N}\right\}$ be a sequence of sets.

Then the limit inferior of the sequence, denoted $\displaystyle \liminf_{n \mathop \to \infty} \ E_n$, is defined as:

$\displaystyle \liminf_{n \mathop \to \infty} \ E_n := \set {x: x \in E_i \text{ for all but finitely many } i}$

## Also denoted as

The limit inferior of $E_n$ can also be seen denoted as:

$\displaystyle {\underline {\lim} }_{n \mathop \to \infty} \ E_n$

but this notation is not used on $\mathsf{Pr} \infty \mathsf{fWiki}$ because it does not render well.

## Also see

• Results about limits inferior of set sequences can be found here.

## Linguistic Note

The plural of limit inferior is limits inferior. This is because limit is the noun and inferior is the adjective qualifying that noun.