This question is related to this post, for which I received a really good answer that gave a beautifull solution, yet I am still trying to understand one thing on the side that is not covered by the answer, which is the following claim:
Suppose we take a Schwartz function $f \in S(\mathbb{R})$ that satisfies $f(x_0) = 0$. Then, we have
\begin{equation} g(x) = (x - x_0)^{-1}f(x) \quad \in S(\mathbb{R}) \quad (x \neq x_0) \end{equation}
I am struggeling to come up with an idea to show this without reverting to the plain definition of a Schwartz function and trying to find the bounds on $g$. But is there another , more immediate answer? From my notes it sounds like there is, but there are no immediate steps.
The answer in the post that I linked this one to actually constructs g specifically, however in my notes the claim is made without any reference to a construction so I am guessing that I miss something here with regards to the properties of g that are imposed on it simply by the equation above... Thanks so much for your help!