Hi

If E is non empty set on a measurable space. and that (A_n)_{n \geq 1} be elements in powerset 2^E.

I wish to show that

lim_n sup B_n = lim_n inf B_n = \bigcup_{n=1} ^{\infty} B_n

this is supposedly only true if B_n \subseteq B_{n+1} for all n \geq 1, and that

lim_n sup B_n = lim_n inf B_n = \bigcap_{n=1} ^{\infty} B_n

which is only true if

B_n \supseteq B_{n+1} for all n \geq 1.


What I know:

What is that lim_n inf B_n = union (all m) of intersections (all n > m) of B_n

and

lim_n sup B_n = intersection (all m) of unions (all n > m) of B_n.

then if lim_n sup B_n = lim_n inf B_n = lim B_n

Could somebody please give me a hint on how to use the above to show that

lim_n sup B_n = lim_n inf B_n = \cup_{n=1} ^\infty B_n

and

lim_n sup B_n = lim_n inf B_n = \cap_{n=1} ^\infty B_n

???

Sincerely Yours
Euroman25