# Math Help - cant understand how they use a theorem question..

1. i marked a place where they use theorem 4

but theorem 4 talks about that if one sequence is bigger then the other
then so is their liminf and limsup

but i cant understand how they use this fact for getting the marked result
what does theorem 4 has to do with the limsup of the sum of sequences
??

why a_n +b_n =limsup(a_n +b_n)
??

2. It's ok, because the limsup of a constant is that constant.
They have $a_n+b_n, where $C=a+b+\epsilon$.
Now take the limit sup of both sides of the equation.
$\limsup a_n+b_n<\limsup C=C$.

And I pointed out last night that $\limsup (a_n+b_n)\le \limsup a_n +\limsup b_n$.
This example shows the inequality...
$a_n=0,1,0,1,0,1...$ and $b_n=1,0,1,0,1,0,...$
So $\limsup (a_n+b_n)=\limsup \{1,1,1,1,1,1...\}=1$
while $\limsup a_n +\limsup b_n=1+1=2$.

3. ok how do i approach this one using the previos method
$
limsup(x_n+y_n)=>liminf x_n+limsup y_n
$

??