Limsup and liminf (sequence of sets)

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Definition

Given a set [ilmath]X[/ilmath] and a sequence [ilmath](A_n)^\infty_{n=1}[/ilmath] of subsets of [ilmath]X[/ilmath], so [ilmath](A_n)^\infty_{n=1}\subseteq\mathcal{P}(X)[/ilmath], we may define the superior limit ([ilmath]\text{Lim sup} [/ilmath]) and inferior limit ([ilmath]\text{Lim inf} [/ilmath]) of [ilmath](A_n)[/ilmath] as follows:

Lim sup

• [math]\mathop{\text{Lim sup} }_{n\rightarrow\infty}(A_n):=\Big\{x\in X\ \Big\vert\vert\{n\in\mathbb{N}\ \vert\ x\in A_n\}\vert=\aleph_0 \Big\}[/math]
  • In words: The superior limit of [ilmath](A_n)[/ilmath] is the set that contains [ilmath]x\in X[/ilmath] given that [ilmath]x[/ilmath] is in (countably) infinitely many elements of the sequence.

Lim inf

• [math]\mathop{\text{Lim inf} }_{n\rightarrow\infty}(A_n):=\Big\{x\in X\ \Big\vert\vert\{n\in\mathbb{N}\ \vert x\notin A_n\}\vert\ne\aleph_0\Big\}[/math]
  • In words: The inferior limit of [ilmath](A_n)[/ilmath] is the set that contains [ilmath]x\in X[/ilmath] given that [ilmath]x[/ilmath] is in all but a finite number of elements of [ilmath](A_n)[/ilmath].

Distinction

One may think to "not be in a finite number of elements" is "to be in an infinite number of elements" and conclude wrongly that these definitions are the same. This is because an element can both be in and not be in an infinite number of elements!

Example

Let [ilmath]a\in X[/ilmath] be some arbitrary point.

• Consider the sequence [ilmath](a_n)_{n=1}^\infty[/ilmath] with [ilmath]A_n:=\left\{\begin{array}{lr}\{a\} & n\text{ is even}\\\emptyset & n\text{ is odd}\end{array}\right.[/ilmath]
  • Now [ilmath]a[/ilmath] is in an infinite number (namely all the even [ilmath]n[/ilmath]s) and not in an infinite number (all the odd [ilmath]n[/ilmath]s) too.
    • [math]\mathop{\text{Lim sup} }_{n\rightarrow\infty}(A_n)=\{a\}[/math] as the number of elements containing [ilmath]a[/ilmath] is (countably) infinite.
    • [math]\mathop{\text{Lim inf} }_{n\rightarrow\infty}(A_n)=\emptyset[/math] as no [ilmath]x\in X[/ilmath] is in an infinite number and only not in a finite number.