Thread: 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 $\displaystyle a_n+b_n<C$, where $\displaystyle C=a+b+\epsilon$.
Now take the limit sup of both sides of the equation.
$\displaystyle \limsup a_n+b_n<\limsup C=C$.

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

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