Summation of a real sequence

While trying to prove a mathematical relationship, I ended up with the following term

$\displaystyle \sum_{k=0}^{\infty} \sum_{n=0}^k kF(n)G(k-n) $

Where, F and G represent functions that accept inter arguments and return real numbers.

Now, the final goal of my proof is equal to

$\displaystyle \sum_{x=0}^{\infty} \sum_{y=0}^{\infty} (x+y)F(x)G(y) $

Can some one please help me if the above two can be proved to be equivalent. I would really appreciate if you could please illustrate all the intermediate steps in this proof.