If trigonometric series is $0$ everywhere, then its coefficients are $0$

Question

How to prove that if $\forall x \in \mathbb R, \sum_{n=-\infty}^\infty c_n e^{inx} = 0$, then $c_n=0$ for all $n$ ?

I feel something could be done by integrating the series, but how to switch the sum and the integral then ? Nothing is assumed on the type of convergence or the regularity of $c_n$ (is it in $\ell^2$).