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I have a problem with the following exercise. I don't really have an idea where to start. I'm glad about every help. So here is the exercise:

Suppose $f\colon R \to R$ is a function in $L^1(R)$ (i.e integrable function). Show that $$\lim_{n \to \infty}\int f(x)\sin(nx)dx=0.$$

Thanks already for any help!

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    Orange you glad Arturo fixed your title?2011-12-14
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    The question does not make sense as written: $\int f(x)\sin(nx)\,dx$ is a family of functions; what does it mean for the family of functions to "go to zero"?2011-12-14
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    Did you mean $\sin( \frac{x}{n})$?2011-12-14
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    obviously $\int_a^b\sin(nx)dx\to0$. now approximate $f$ by simple functions (see http://planetmath.org/encyclopedia/RiemannLebesgueLemma.html)2011-12-14

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