Let be differentiable at and let and be two sequences converging to such that for . Prove that
The question doesn't allow us to use mean value theorem directly to the function.
Anyway,the hint given to this question is that using , show that this lie between and .
I've tried using this hint, but I can't get anywhere between and . And if I'm not wrong, we can't assume that the function is monotonic as there might be cases where for some and for the remaining integers in .
So any idea on how to continue from the hint above?
Thanks in advance.