Lim sup of sequence of sets and theirs unions I have to prove the following equality: 
$$\left(\limsup_{n \to \infty} A_n\right)\cup\left(\limsup_{n \to \infty} B_n\right) = \limsup_{n \to \infty} \left(A_n\cup B_n\right)$$
Can somebody help me to prove this?

If $(C_n,n\geqslant 1)$ is a collection of sets, then $x\in\limsup_n C_n$ means that $I(x):=\{n\geqslant 1,x\in C_n\}$ is infinite. 
Since $A_n\subset A_n\cup B_n$ for each $n$, we certainly have $\limsup_nA_n\subset\limsup_n (A_n\cup B_n)$. 
For the reverse inclusion, let $x\in \limsup_n (A_n\cup B_n)$. Then $x\in A_n\cup B_n$ for infinitely many $n$. Defining $I:=\{n\geqslant 1,x\in A_n\}$ and $J:=\{n\geqslant 1,x\in B_n\}$, we have that $I\cup J$ is infinite, hence $I$ or $J$ is infinite. 
