This is what I tried:Let and be three sequences of real numbers.

Suppose that for every n, and let . Assuming that limsup a_n and limsup b_n are finite, prove the following inequality:

Assume

Therefore

This is only true if eventually or eventually. Is there a way to adapt this proof for sequences which constantly oscillate between or ? (ie.

I'm not really sure what to put for this part of the question. It seems like nothing conclusive can be written.What can you say about the value of as compared to and ?

Justify your answer.

Does anyone have any ideas?