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.