What is the Relationship Between Liminf and Limsup in Subset Sequences?

[SpaceWalrus]

I have been told that given a sequence of subsets, liminf $A_n\subseteq$ limsup $A_n$. I have spent some time trying to cook up such a sequence, but I can only ever show they are equal. Can someone point me in the right direction?

 

[mathman]

If there is no restriction on A_n, then you could set up a sequence:
A_n = empty set for even n and A_n = whole space for odd n.

 

[SpaceWalrus]

If there is no restriction on A_n, then you could set up a sequence:
A_n = empty set for even n and A_n = whole space for odd n.

I don't know if that one works. Note that

liminf $A_n = A_1 \cup [A_1 \cap A_2] \cup [A_1 \cap A_2 \cap A_3] \dots = X \cup [X \cap \emptyset] \cup [X \cap \emptyset \cap X] \dots = X \cup \emptyset \cup \emptyset \dots = X,$

while

limsup$A_n = A_1 \cap [A_1 \cup A_2] \cap [A_1 \cup A_2 \cup A_3] \dots = X\cap [X \cup \emptyset] \cap [X \cup \emptyset \cup X] \dots = X \cap X \cap X \dots = X.$

Hence they are equal.

 

[mathman]

I am a little confused by your definitions. My understanding is that limsup consists of all points which are in an infinite number of sets of the sequence, which would be the whole space. Liminf according to my understanding are those points which belong to all but a finite number of sets in the sequence, which would be none.

To use your expression I think you need to start with A_n (not A₁) and then let n become infinite.

 

[SpaceWalrus]

Yeah, I had the definition wrong. UGH! Thanks!

 

[dugale]

How can I prove that liminf of a sequence of events An is a subset of lim sup of An given n goes to infinity? I'm inexperienced in constructing proofs, so please do not be strict.

 

[mathman]

One way of getting there is to look at
sup_n = A_n∪A_n+1∪...
inf_n = A_n∩A_n+1∩...
sup_n contains inf_n

So let n become infinite and the relationship still holds.

 

[dugale]

Thank you, it's clear now.