I'm reading the book "Malliavin Calculus for Lévy Processes with Applications to Finance".
At one point the authors prove that the Leibniz rule holds for the Malliavin Derivative $D_t$ taken at functions in $\mathbb{D}_{1,2}^0$, which is the subset of $L^2(P)$ consisting of those functions with only finitely many nonzero terms in their Wiener-Itô Chaos expansion.
It is clear that if $F\in\mathbb{D}_{1,2}^0$, then $F\in\mathbb{D}_{1,2}$, which is the space of functions such that $\sum_{n=1}^{\infty} nn!\|f_n\|_{L^2([0,T]^n)}^2<\infty$, where $\sum_{n=0}^{\infty} I_n(f_n)$ is their chaos expansion.
Now they state that if $F_1,F_2\in\mathbb{D}_{1,2}^0$, then $F_1\cdot F_2\in L^2(P)$, and they say this is because the Gaussian random variables have all finite moments.
This doesn't seem trivial to me. How do I derive this from the fact that Gaussian random variables have all finite moments?