Let $A$ be an open subset of $\mathbb{R}^n$. For an infinitely differentiable function $f(x)$ on A$, define
$$[f](\rho) = \sum_{ \alpha \in \mathbb{N}^n} { \frac{\sup_{x \in A} | \partial_x^{\alpha} f(x) |}{|\alpha|!} \rho^{|\alpha|} }.$$
Is it true that $[fg](\rho) \ll [f](\rho) \cdot [g](\rho)$? (That is, are the coefficients of $[fg]$ less than or equal to the corresponding coefficients of $[f] \cdot [g]$?)
I know that by Leibniz' rule, we have
\begin{align} [fg](\rho) & = \sum_{ \alpha \in \mathbb{N}^n} { \frac{\sup_{x \in A} | \partial_x^{\alpha} fg(x) |}{|\alpha|!} \rho^{|\alpha|} } \\ & \leq \sum_{ \alpha \in \mathbb{N}^n} { \frac{ \sum_{\beta + \gamma = \alpha} \frac{\alpha!}{\beta! \gamma!} \sup_{x \in A} | \partial_x^{\beta} f(x) | \cdot \sup_{x \in A} | \partial_x^{\gamma} g(x) |}{|\alpha|!} \rho^{|\alpha|} } \\ & \leq \sum_{ \alpha \in \mathbb{N}^n} { \frac{1}{\binom{|\alpha|}{\alpha}} \sum_{\beta + \gamma = \alpha} \frac{1}{\beta! \gamma!} \sup_{x \in A} | \partial_x^{\beta} f(x) | \cdot \sup_{x \in A} | \partial_x^{\gamma} g(x) | \rho^{|\alpha|} }. \end{align}
But $$ \frac{1}{\beta!} \leq n^{|\beta|} \frac{1}{|\beta|!} \text{ and } \frac{1}{\gamma!} \leq n^{|\gamma|} \frac{1}{|\gamma|!}$$ and I have problem with those $n^{|\beta|}$ and $n^{|\gamma|}$. Am I missing something here? Did I make a mistake somewhere?