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Is there an easy proof of the following fact?

Let $a_0, \ldots, a_n, b_1, \ldots, b_n$ be real numbers, not all zero. Then, the function $$a_0 + a_1 \cos x + b_1 \sin x + a_2\cos 2x+b_2\sin 2x+\ldots+ a_n \cos nx + b_n \sin nx$$ has at most $2n$ roots in the range $0\le x<2\pi$.

(The above expression is a Fourier series truncated to the first $2n+1$ terms.)

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migrated from mathoverflow.net Jan 25 '14 at 22:20

This question came from our site for professional mathematicians.

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    $\begingroup$ That is quite obvious if you express your sines and cosines in term of $e^{ix}$. But your question is not appropriate for MO, it is definitely not at research level. $\endgroup$ – abx Jan 25 '14 at 19:23
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    $\begingroup$ abx, thanks for the hint. Though, after re-reading the FAQ, I still think my question is appropriate. It arose from reading a graduate-level book (Ziegler's "Lectures on Polytopes"), and the FAQ says questions don't have to be about new mathematics. Is it a bad thing that the answer is easier than expected? $\endgroup$ – Gabriel Nivasch Jan 25 '14 at 20:09

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