# Real analysis

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.

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

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

and

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.