I have a function $f\in \mathcal{S}$ (i.e of Schwartz class), and I want to show there exist constants $C,k>0$ s.t $\|f\|_p \leq C(\sup_{x\in \mathbb{R}} |f(x)| + \sup_{\mathbb{x\in \mathbb{R}}} |x^k f(x)|)$ for every $ p \in [1,\infty]$.
For $p=1,\infty$ it's obvious from definition, I mean I can take f with compact support and this will prove for $p=1$, for $p=\infty$ it's trivial.
But for $ p \in (1,\infty)$ I find myself at a mess, I need to do integration by parts inductively but I don't seem to find the right approach, I guess I need to use here Leibnitz general product rule, but I don't see how to come to suitable constants.
Any help , is appreciated.