Let a<b. Let f_n: [a,b] -> R be a sequence of functions such that, for each n in N ( N set of natural numbers),f_n is differentiable on (a,b). Suppose that for all n in N,
Sup on [a,b] of | f'_n(x) | < or = to M, where M is in R. ( Sup is supremum = least upper bound)
Prove that for all n in N and all x, y in [a,b], one has that,
|f_n(y) - f_n(x) | < or = to M|y - x|. Justify every step please.
Since Sup|f'_n(x)|<=M, then for any n and any x on [a,b], we ...
A Sequence of Functions and the Mean Value Theorem are investigated.