Hi there, I'm self-studying Protter's "A First Course In Real Analysis" and I've come across a limit problem that I've tried hard to solve, but yet have had difficulties completing the proof. The theorem is as stated:

Theorem: Suppose that and . If there is a number such that for then

To prove this we need (I assume) to show that such that .

After trying to prove this directly, I come up with , where is the positive number used in the argument of and but obviously ,

and so this conclusion does not hold for all A>0 .

Again, I can show my work if you would like (I'm clumsy with latex so I just stated my result), but any help would be appreciated. Thanks a lot.

Eric