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    Sequence of Functions and Mean Value Theorem

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    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.

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    Since Sup|f'_n(x)|<=M, then for any n and any x on [a,b], we ...

    Solution Summary

    A Sequence of Functions and the Mean Value Theorem are investigated.