It's ok, because the limsup of a constant is that constant.
They have , where .
Now take the limit sup of both sides of the equation.
And I pointed out last night that .
This example shows the inequality...
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)