Prove there is only finitely many Fourier coefficients of f that are non-zero

Question

Can you give me a proof of this?

Answer 1

Another way: $f$ being a finite Fourier sum is analytic. But an analytic function on ${\mathbb R}$ can only have isolated zeros.

Answer 2

Hint: $f$ is the sum of its Fourier series, and therefore if there are only finitely many nonzero coefficients it is a trigonometric polynomial. Trigonometric polynomials are rational functions in $e^{ix}$ and therefore ...