Proving Limsup of AUB = Limsup(A) U Limsup(B)

[demZ]

Homework Statement

Given A = {A_n} and B = {B_n}. Prove that Limsup ({AUB}) = Limsup(A) U Limsup(B).

Homework Equations

The Attempt at a Solution

The result of both sides is a set, so I attempted to use double inclusion to prove it. Used the definition of limsup, but didn't manage to prove that for x in limsup(AUB), it is either in limsup(A) or limsup(B) and vice versa.

 

[jbunniii]

Do you know a formula for the lim sup of a sequence of sets, expressed in terms of unions and intersections?

 

[micromass]

You could use the following:
x is in limsup A if and only if x is in An for infinitely many n.

Then x is in \limsup{A\cup B}, if it is in infinitely many A_n\cup B_n. But then it must of course be in infinitely many An or Bn; thus x is in \limsup A_n\cup \limsup B_n.
The other implication is the same thing...