# Proof for multiplying generating functions

I've learned that multiplying two generating functions f(x) and g(x) gives you the result

\begin{equation*} \sum_{k=0}^\infty\left(\sum_{j=0}^k a_j\,b_{k-j}\right)x^k. \end{equation*}

I've used the result, but it was presented in my class without proof and I'm having some trouble tracking one down. Weak google-foo today, I suppose. Can anyone give me a pointer to a proof? If this is a question better answered in book form, that is fine as well.

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$\sum_{n=0}^{\inf} a_n z^n \cdot \sum_{n=0}^{\inf} b_n z^n = (a_0+a_1z+a_2z^2+ \cdots)(b_0+b_1z+b_2z^2+ \cdots)$
$=a_0b_0+(a_0b_1 + a_1b_0)z + (a_0b_2+a_1b_1+a_2b_0)z^2 + \cdots$
$=c_0+c_1z+c_2z^2+ \cdots$
where $c_n=\sum_{k=0}^n a_k b_{n-k}$