Q: Average of limsup and liminf Suppose $(a_n)$ and $(b_n)$ are sequences of reals in $[0,1]$. Is it true that 
$\frac{\limsup_n (a_n+b_n) + \liminf_n (a_n+b_n)}{2} = \frac{\limsup_n a_n + \liminf_n a_n}{2} + \frac{\limsup_n b_n + \liminf_n b_n}{2}$?
I don't think its true.. but I can't find a counterexample. I know its true if either $(a_n)$ or $(b_n)$ converges. 

A: Here's a counterexample: 
$a_{3k}=1$ and $a_{3k+1}=-1$ and $a_{3k+2}=1$.
$b_{3k}=1$ and $b_{3k+1}=1$, $b_{3k+2}=-1$.
Then 


*

*$\limsup$ of either is $1$ and $\liminf$ of either is $-1$. Thus, RHS is zero. 

*As for LHS,  $\limsup a_n+b_n =2$ and  $\liminf a_n+b_n = 0$. Therefore LHS is $1$. 

