Cant understand how they use a theorem question

[transgalactic]

http://www.freeimagehosting.net/uploads/th.38f39915df.gif

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
it doesn't tell anything about the limsup of the sum of the sequences i can't 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
??how they made that step?

 

[wywong]

Let $$\alpha_n = a_n+b_n,\ \beta_n = a+b+\epsilon$$(constant) and apply theorem 4 for the sequences $$\alpha_n$$ and $$\beta_n$$ and you will get the result.

 

[transgalactic]

i can't see how i get the supreme of the sum of both sequences
using theorem 4

??

 

[transgalactic]

theorem 4 talks about that if one sequence is bigger then the other
then so is the liminf of the first is bigger then the other and so is for limsup
it doesn't tell anything about the limsup of the sum of the sequences
??

 

[transgalactic]

Let $$\alpha_n = a_n+b_n,\ \beta_n = a+b+\epsilon$$(constant) and apply theorem 4 for the sequences $$\alpha_n$$ and $$\beta_n$$ and you will get the result.

i can't understand what you are saying
how ill get the result
/??

 

[transgalactic]

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