Cauchy product associativity proof Great. I need a proof that the Cauchy product is an associative operation. I can easily proof that it is a commutative operation, find identity series and find invertible and inverse series, BUT for some reason I fail to proof this damn associativity.
The proof should not use any fancy theorems or so... Rather, it should be a simple algebraic proof that $\forall_{n\ge n_0}\sum^n_{k=n_0}\sum^k_{l=n_0}a_lb_{k-l}c_{n-k}=\sum^n_{n=n_0}\sum^k_{l=n_0}a_{n-k}b_lc_{k-l}$.
Now I'm really sorry for this dumb question. Feel free to down-vote it as hard as you please, but could you kindly answer it? Thanks.

(Assuming you work with coefficients $\in\mathbb R$ or similar) The Cauchy product of the coefficients of polynomials $A,B$ gives us the coefficients of the unique(!) polynomial $P$ with $P(x)=A(x)B(x)$ for all $x$. Since $\bigl(A(x)B(x)\bigr)C(x) = A(x)\bigl(B(x)C(x)\bigr)$ for all $x$, both three-fold Cauchy products obtained must give the same result.
