Well the problem reads: Suppose

and

exists for every sequence

(where n goes from

) in

such that

. Prove that

exists.

I'm going to assume that

does not exist and prove it by contradiction. So then we still have

. But since the sequence

lies in the interval

the

can't exist because we assumed that right hand limit:

does not exist, so the limit as anything approaches a in the interval

can't exist. But that is a contradiction since we are given

.

I'm not sure if this is right because it seems a little short. Can someone let me know if this is the right approach? Thanks, Chad.

Edit: Maybe I have to use Bolzano-Weierstrass and say that

has a convergent sub sequence that tends to a as x approaches infinity, not really sure that makes a difference though since we already have a sequence that converges to a.