I am seeing the Schwartz Space for the first time today and I have trouble understanding the following argument: Given that $f \in S({\mathbb{R}})$ with $f(x_0) = 0$ Taylor's theorem tells us that we can find a function $g \in C^\infty(\mathbb{R})$ such that \begin{equation} f(x) = g(x)(x - x_0) \end{equation} Hence, for $x \neq x_0$ we can write $g(x)$ as \begin{equation} g(x) = (x - x_0)^{-1} f(x) \end{equation} According to my notes, from this it should be clear that $g \in S(\mathbb{R})$, but how can I see this immediately ? In order to derive this I think would need to direvtly go via the definition, i.e. show for each $k,m = 0,1,2, \dots$ there exist constants $c_{k,m}$ such that \begin{equation} \sup_x |x^k g^{(m)}(x)| \leq c_{k,m} \end{equation} or is there a shorter way to infer this ? By "shorter" I mean for example a statement that somehow the constants that bound $f$ are also bounds on $g$, probably with some amendment.
Edit: Does it maybe have to do with the fact that $f$, being smooth, has a Tayler expansion and so \begin{equation} g(x) = \sum^\infty_{n = 1} \frac{f^{(n)}(x_0)}{n!}(x - x_0)^{n-1} \quad (\text{note } f(x_0)) = 0 \end{equation}
Thanks for your help!