Nah, no need for contradiction. If , then , where for all n, which must go to some positive value (C) since B is finite. Then
But we also know this limit is B, therefore B = A + C, and thus .
How do you know B is finite? Also, how do you know the c sub n has limit C?
In class, we have done these type of proof using the definition of converge. I can follow this, mostly, but I don't think I'm allowed to do the proof this way.
If a sequence has a limiting value of B, then B is a number and is clearly finite.
In order for all the b terms to be greater than the a terms, then it's the same as adding some value to the a terms. Since the a terms go to some value A, in order for the b terms to also go to some value, then the remaining c terms also have to go to some value.
This is helpful, but did you let epsilon = -L/2? which is positive because L is less than zero. But where did you get -L/2? I'm guessing it's from some scratch work that should be obvious to me, but isn't.
Also, we haven't proved or used that the limit of the difference two convergent sequences is the the difference of the limits. I know it's true, but since we haven't proved it I can't use it.