Definition:Limit Inferior of Sequence of Sets

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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:

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


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.