Proof of Properties of Fourier series in CT

Question

I feel problem in understanding the proof of Fourier series properties
1) Time scaling
$$b_k = \frac{1}{T}\int_{T}x(t)e^{jk\omega_0t}dt$$ a= scaling factor $$ = \frac{a}{T}\int_{T/a}x(at)e^{jk(\omega_0a)t}dt$$ L=at $$ = \frac{a}{T}\int_{T}x(L)e^{jk\omega_0L}\frac{dt}{a}$$ $$ = \frac{1}{T}\int_{T}x(L)e^{jk\omega_0L}dt = a_k$$ Q) For this I want to ask that how $T/a$ term in the integral changes $dt/a$?
2)Multiplication $$x(t)y(t)= \sum^{+\infty}_{k\ =\ -\infty} a_k e^{jk\omega_0t}dt\sum^{+\infty}_{k\ =\ -\infty} b_k e^{jl\omega_0t}dt$$ $$ \sum^{+\infty}_{k\ =\ -\infty} \sum^{+\infty}_{l\ =\ -\infty} a_k b_le^{j(k+l)\omega_0t}$$ let m=k+l $$ \sum^{+\infty}_{m\ =\ -\infty} [\sum^{+\infty}_{l\ =\ -\infty} a_{m-l} b_l]e^{jm\omega_0t} $$ Q) I couldn't understand this last step, why the summation changes from $k$ to only $m$? isn't it should be $m-l$ and also is it done here to prove the multiplication property? or there are more steps? because in the Signals and Systems 2nd Edition Chapter 3, page 204 the just mentioned that $$ x(t)y(t) -> h_k= \sum^{+\infty}_{l\ =\ -\infty}a_lb_{k-l}$$