Editing Set-theoretic limit (section)

==Definitions==

===The two definitions===
Suppose that <math>\left(A_n\right)_{n=1}^\infty</math> is a sequence of sets. The two equivalent definitions are as follows.

* Using [[Union (set theory)|union]] and [[Intersection (set theory)|intersection]]: define<ref name="probpath">{{cite book| last1=Resnick|first1=Sidney I.|title=A Probability Path|date=1998|publisher=Birkhäuser|location=Boston|isbn=3-7643-4055-X}}</ref><ref>{{Cite book |last=Gut |first=Allan |url=https://link.springer.com/10.1007/978-1-4614-4708-5 |title=Probability: A Graduate Course: A Graduate Course |date=2013 |publisher=Springer New York |isbn=978-1-4614-4707-8 |series=Springer Texts in Statistics |volume=75 |location=New York, NY |language=en |doi=10.1007/978-1-4614-4708-5}}</ref> <math display="block">\liminf_{n \to \infty} A_n = \bigcup_{n \geq 1} \bigcap_{j \geq n} A_j</math> and <math display="block">\limsup_{n \to \infty} A_n = \bigcap_{n \geq 1} \bigcup_{j \geq n} A_j</math> If these two sets are equal, then the set-theoretic limit of the sequence <math>A_n</math> exists and is equal to that common set. Either set as described above can be used to get the limit, and there may be other means to get the limit as well.
* Using [[indicator function]]s: let <math>\mathbb{1}_{A_n}(x)</math> equal <math>1</math> if <math>x \in A_n,</math> and <math>0</math> otherwise. Define<ref name="probpath"/> <math display=block>\liminf_{n \to \infty} A_n = \Bigl\{ x \in X : \liminf_{n \to \infty} \mathbb{1}_{A_n}(x) = 1 \Bigr\}</math> and <math display=block>\limsup_{n \to \infty} A_n = \Bigl\{ x \in X : \limsup_{n \to \infty} \mathbb{1}_{A_n}(x) = 1 \Bigr\},</math> where the expressions inside the brackets on the right are, respectively, the [[limit infimum]] and [[limit supremum]] of the real-valued sequence <math>\mathbb{1}_{A_n}(x).</math> Again, if these two sets are equal, then the set-theoretic limit of the sequence <math>A_n</math> exists and is equal to that common set, and either set as described above can be used to get the limit.

To see the equivalence of the definitions, consider the limit infimum. The use of [[De Morgan's law]] below explains why this suffices for the limit supremum. Since indicator functions take only values <math>0</math> and <math>1,</math> <math>\liminf_{n \to \infty} \mathbb{1}_{A_n}(x) = 1</math> if and only if <math>\mathbb{1}_{A_n}(x)</math> takes value <math>0</math> only finitely many times. Equivalently, 
<math display=inline>x \in \bigcup_{n \geq 1} \bigcap_{j \geq n} A_j</math> 
if and only if there exists <math>n</math> such that the element is in <math>A_m</math> for every <math>m \geq n,</math> which is to say if and only if <math>x \not\in A_n</math> for only finitely many <math>n.</math> 
Therefore, <math>x</math> is in the <math>\liminf_{n \to \infty} A_n</math> if and only if <math>x</math> is in all but finitely many <math>A_n.</math> For this reason, a shorthand phrase for the limit infimum is "<math>x</math> is in <math>A_n</math> all but finitely often", typically expressed by writing "<math>A_n</math> a.b.f.o.".

Similarly, an element <math>x</math> is in the limit supremum if, no matter how large <math>n</math> is, there exists <math>m \geq n</math> such that the element is in <math>A_m.</math> That is, <math>x</math> is in the limit supremum if and only if <math>x</math> is in infinitely many <math>A_n.</math> For this reason, a shorthand phrase for the limit supremum is "<math>x</math> is in <math>A_n</math> infinitely often", typically expressed by writing "<math>A_n</math> i.o.".

To put it another way, the limit infimum consists of elements that "eventually stay forever" (are in {{em|each}} set after {{em|some}} <math>n</math>), while the limit supremum consists of elements that "never leave forever" (are in {{em|some}} set after {{em|each}} <math>n</math>). Or more formally:
:{|
|-
|<math Display="inline">\lim_{n\in\N}A_n = L \quad \Longleftrightarrow</math> &nbsp; &nbsp; || for every <math>x\in L</math> &nbsp; &nbsp; &thinsp; there is a <math>n_0\in\N</math> with <math>x\in A_n</math> for all <math>n\ge n_0</math> and
|-
| ||for every <math>y\in X\!\setminus\! L</math> there is a <math>p_0\in\N</math> with <math>y\not\in A_p</math> for all <math>p\ge p_0</math>.
|}

===Monotone sequences===
{{anchor}}
The sequence <math>\left(A_n\right)</math> is said to be '''nonincreasing''' if <math>A_{n+1} \subseteq A_n</math> for each <math>n,</math> and '''nondecreasing''' if <math>A_n \subseteq A_{n+1}</math> for each <math>n.</math> In each of these cases the set limit exists. Consider, for example, a nonincreasing sequence <math>\left(A_n\right).</math> Then 
<math display=block>\bigcap_{j \geq n} A_j = \bigcap_{j \geq 1} A_j \text{ and } \bigcup_{j \geq n} A_j = A_n.</math>
From these it follows that
<math display=block>\liminf_{n \to \infty} A_n = \bigcup_{n \geq 1} \bigcap_{j \geq n} A_j = \bigcap_{j \geq 1} A_j = \bigcap_{n \geq 1} \bigcup_{j \geq n} A_j = \limsup_{n \to \infty} A_n.</math>
Similarly, if <math>\left(A_n\right)</math> is nondecreasing then 
<math display=block>\lim_{n \to \infty} A_n = \bigcup_{j \geq 1} A_j.</math>

The [[Cantor set#Construction and formula of the ternary set|Cantor set]] is defined this way.