# Definition:Limit Inferior of Sequence of Sets

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## Definition

### Definition 1

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

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

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

### Definition 2

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

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

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

## 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.

Some sources merely present this as:

$\ds \underline \lim E_n$

## 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.