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)

??