Proof of properties of Fourier series in CT
Looking at the change of terms in the equality:
$$\frac{a}{T}\int_{T/a}x(at)e^{jk(\omega_0a)t}dt = \frac{a}{T}\int_{T}x(L)e^{jk\omega_0L}\frac{dL}{a}$$
$$at = L\implies \begin{cases}\text{if }t=T/a\implies L=T\\adt=dL\implies dt=\frac{dL}{a}\end{cases}$$
Looking at the equality again:
$$ \sum^{+\infty}_{k\ =\ -\infty} \sum^{+\infty}_{l\ =\ -\infty} a_k b_le^{j(k+l)\omega_0t}=\sum^{+\infty}_{m\ =\ -\infty} \left[\sum^{+\infty}_{l\ =\ -\infty} a_{m-l} b_l\right]e^{jm\omega_0t} $$
$$m=k+l\implies \begin{cases}k=m-l\quad({\scriptstyle{\rm indeed}})\\\text{and } m-l=\pm \infty \implies  m=\pm\infty+l=\pm\infty\end{cases}$$