We know that if you are multiplying fps $A$ by a fps $B$, the product can be written as: $$AB = \sum_{i=0}^{\infty}\left(\sum_{j=0}^ia_jb_{i-j}\right)X^i$$
or $$a_0b_0 + (a_0b_1 + a_1b_0)x + (a_0b_2 + a_1b_1 + a_2b_0)x^2 + \ldots$$
so each of $(a_0b_0), (a_0b_1 + a_1b_0), (a_0b_2 + a_1b_1 + a_2b_0), \ldots$ must equal $0$.
The only way for this to happen is if $a_n = 0$, for all n or $b_n = 0$, for all $n$.
How do I prove this? Induction?
Assume neither $A=0$ nor $B=0$. Let $n$ be minimal with $a_n\ne 0$, let $m$ be minimal with $b_m\ne 0$. Then the coefficient of $x^{n+m}$ in the Cauchy product equals $a_nb_m\ne 0$.