# Real Analysis Problem

We have just finished up integration and are done with a first course in analysis, so 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

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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... this laying around:

"Let f(x) be absolutely integrable on [a,b]. Then, for any epsilon > 0, there exists a continuous, piecewise smooth function g(x) such that

*S* | f(x) - g(x) | dx <= epsilon (using your *S* notation)

Anyhow... here's the proof for when f is absolutely integrable.

Let epsilon be an arbitrarily small positive number. By the lemma above, there exists a ...