Proof of properties of Fourier series in CT I feel problem in understanding the proof of Fourier series properties  


Time scaling
\begin{align}
b_k &amp;= \frac{1}{T}\int_{T}x(t)e^{jk\omega_0t}dt\\
&amp; = \frac{a}{T}\int_{T/a}x(at)e^{jk(\omega_0a)t}dt\tag{$\scriptstyle{a - \text{scaling factor}}$}\\
&amp; = \frac{a}{T}\int_{T}x(L)e^{jk\omega_0L}\frac{dL}{a}\tag{$\scriptstyle{L=at}$}\\
&amp;= \frac{1}{T}\int_{T}x(L)e^{jk\omega_0L}dL = a_k\\
\end{align}
For this I want to ask that how $T/a$ term in the integral changes to $T$, and $dt$ changes to $dL/a$?  
Multiplication
\begin{align}
x(t)y(t)&amp;= \sum^{+\infty}_{k\ =\ -\infty} a_k  e^{jk\omega_0t}\sum^{+\infty}_{k\ =\ -\infty} b_k  e^{jl\omega_0t}\\ &amp;=\sum^{+\infty}_{k\ =\ -\infty} \sum^{+\infty}_{l\ =\ -\infty} a_k b_le^{j(k+l)\omega_0t}\\
&amp;=\sum^{+\infty}_{m\ =\ -\infty} \left[\sum^{+\infty}_{l\ =\ -\infty} a_{m-l} b_l\right]e^{jm\omega_0t}\tag{$\scriptstyle{m=k+l}$}
\end{align}
I couldn't understand this last step, why the summation changes from $k$ to only  $m$? Shouldn't it be $m-l$ and also is it done here to prove the multiplication property? or there are more steps?
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:
$$ \sum^{+\infty}_{k\ =\ -\infty} \sum^{+\infty}_{l\ =\ -\infty} a_k b_le^{j(k+l)\omega_0t}=\sum^{+\infty}_{m\ =\ -\infty} \left[\sum^{+\infty}_{l\ =\ -\infty} a_{m-l} b_l\right]e^{jm\omega_0t} $$
$$m=k+l\implies \begin{cases}k=m-l\quad({\scriptstyle{\rm indeed}})\\\text{and } m-l=\pm \infty \implies  m=\pm\infty+l=\pm\infty\end{cases}$$