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I encountered this problem while studying for an analysis exam. Here is a related question I asked some days ago.
The problem is as follows: Suppose $a_n$ is a decreasing sequence of positive real numbers and that$\sum_{n = 0}^{\infty}{a_n \sin{(nx)}}$ converges uniformly on $\mathbb{R}$, show that $\lim_{n \to \infty}{(n a_n)} = 0.$ Any tip or solution is welcome, and also avoid using Fourier series, because they haven't been introduced in the book so it can be solved without using them.

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    @Peláez: No harm done. (English is not my native language either...)2011-03-26

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$\sum_{i=[(k+1)/2]}^k a_i \sin(ix)$ goes uniformly to 0 as $k\to\infty$. Set $x=\pi/(2k)$. Then all $a_i$'s are $\geq a_k$, all the sines are $\geq1/\sqrt{2}$, hence the sum is $\geq (k-1)/2\times a_k/\sqrt{2}$. Since this goes to 0, so does $k a_k$

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    @user8268 could you explain your answer please? Explain the motivation for letting $i = \frac{k+1}{2}$, how $a_i \geq a_k$, etc. Thanks2015-02-25