Understanding this derivation for Rayleigh's Identity

Question

I'm having trouble understanding this proof of Rayleigh's Identity. More specifically, I'm not too comfortable with the summation notation. Firstly, I'm assuming that $$ \displaystyle\sum_ i \sum_ j =\displaystyle\sum_ {i,j}$$ If this assumption is true, then I'm not following the proof. It seems you can just take the product like so: $$ \displaystyle\sum_ i^ \infty x_ i \sum_ j^ \infty x_ j = \displaystyle\sum_ {i,j}^ \infty x_ i x_ j $$ Can you really do this and just multiply the sums like that? Isn't this incorrect since for example $$ \displaystyle\sum_ {i=1}^ 2 x_ i \sum_ {j=1}^ 2 x_ j = x_ 1^ 2 +2x_ 1 x_ 2 + x_2 ^ 2 $$ where as $$ \displaystyle\sum_ {i,j=1}^ 2 x_ i x_ j = x_ 1^ 2 + x_ 2^ 2 $$ What am I doing wrong?