We say that a real function $f$ of class $\mathcal{C}^{\infty}$ on $\mathbb{R}$ has rapid decay if
$$\forall (n,m) \in \mathbb{N}^2, \lim_{x \rightarrow +\infty} x^m f^{(n)}(x) = \lim_{x \rightarrow -\infty} x^m f^{(n)}(x) = 0$$
We denote $\mathcal{S}$ the set of functions from $\mathbb{R}$ to $\mathbb{R}$ of class $\mathcal{C}^{\infty}$ with rapid decay.
Show that if $P$ is a polynomial function and if $f$ is in $\mathcal{S}$, then $Pf$ belongs to $\mathcal{S}$.