View source for Limit of increasing sequence of sets

==Definition==
Given an [[increasing sequence of sets]] {{M|1=(A_n)_{n=1}^\infty}}, we define the [[limit]] of the sequence as follows<ref>https://proofwiki.org/wiki/Definition:Limit_of_Increasing_Sequence_of_Sets - taken from book I left at home! - Measures, integrals and Martingales</ref><ref name="PAS">Probability and Stochastics - Erhan Cinlar</ref>:
* {{MM|1=\lim_{n\rightarrow\infty}(A_n)=A}} where {{M|A}} is ''the limit of {{M|A_n}}'', and {{MM|1=A:=\bigcup_{n=1}^\infty A_n}}
This may be written as:
* {{M|A_n\uparrow A}}<ref>Measures, Integrals and Martingales - CHECK THIS REF</ref>
** I do not like this notation, as {{M|\uparrow}} only shows the notion of increasing, I prefer {{M|\nearrow}} as this 'combines' (in a very vector-like sense) the {{M|\rightarrow}} of limit and {{M|\uparrow}} of increasing.
* {{M|A_n\nearrow A}}<ref name="PAS"/>
** I prefer this notation, however I always explicitly write {{MM|1=\lim_{n\rightarrow\infty}(A_n)=A}} myself, after letting {{M|1=(A_n)_{n=1}^\infty}} be an increasing sequence.

==Alec's definition==
When I encountered this in a book(<ref name="PAS"/>) I didn't read on and formulated myself the definition of {{M|1=\lim_{n\rightarrow\infty}(A_n)=A}} if:
* {{MM|1=\forall x\in A\exists N\in\mathbb{N}\forall n\in\mathbb{N}[n> N\implies x\in A_n]}} and
* {{MM|1=\forall n\in\mathbb{N}[A_n\subseteq A]}}
Notice the first one alone is insufficient as any subset of some {{M|A_n}} will satisfy it, so I formulated the second. The first also contains the increasing sequence idea as it requires after a certain index all sets contain {{M|x}}.

{{Begin Theorem}}
Proof that {{MM|1=\lim_{n\rightarrow\infty}(A_n)=A:=\bigcup_{i=1}^\infty A_n\iff }} those two conditions. 
{{Begin Proof}}
{{Todo|Be bothered, I've done this on some anonymous notepad, it isn't difficult}}
{{End Proof}}{{End Theorem}}
==References==
<references/>
{{Definition|Set Theory|Measure Theory|Real Analysis}}