Proof of the convolution property of Fourier Series in continuous time
HINT:

Show that with $\Omega_0=2\pi/T$

$$\int_T e^{j\Omega_0(n-p)\tau} d\tau=\begin{cases}T,&amp;\quad n=p\\0,&amp;\quad\text{otherwise}\end{cases}$$

So the sum over $p$ reduces to a single element $Td_ne^{j\Omega_0 nt}$, and the result follows.