Sum of product of Fourier series

This is a consequence of the Dominated Convergence Theorem, see This question.

In details:
$$
sum_{wneq0}sum_{s} a_s a_{s+Nw}
$$
is better written
$$
sum_{s}sum_{z} b_{s,z}^N
$$
where
$$
b_{s,z}^N = a_{s}a_{z} mbox{ when } z=s mbox{ mod } N,, Nneq0 mbox{ and } b_{s,z}^N=0 mbox{ otherwise.}
$$
Now, $|b_{s,z}^N|leq |a_{s}||a_{z}|$ for all $N$, and $$sumsum|a_{s}||a_{z}|=(sum |a_{s}|)^2<infty.$$
On the other hand, for any fixed $z, s$
$$
lim_{Nto infty} b_{s,z}^N = 0 ,(mbox{ since } a_{s+Nw} to 0 mbox{ with }N)
$$
Now apply the DCT to conclude.