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    Real analysis

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    We have just finished up integration and are done with a first course in analysis, chapters 1-6 of Rudin. We are also using the Ross and Morrey/Protter book.

    Please answer question fully and clearly explaining every step. Any solution short of perfect is useless to me. So if you are not 100% sure whether your answer is right, then please do not answer.

    The same problem is also attached as a word document with all the symbols.

    Let f: [a,b] --> R an integrable function. Prove that:

    i) lim *S* f(x) cos(nx) dx = 0


    ii) lim *S* f(x) sin(nx) dx = 0.

    where lim is n as it approaches plus infinity (it is
    not specified so I believe the default when only n
    is listed is to plus infinity, I may be mistaken),

    *S* is my notation for the integral taken from a to b.

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    Solution Summary

    This is a proof regarding limits of integrable functions.