Prove the Schwarz space is contained in $C_0(R^n)$

Question

In our definition,

$$C_0(R^n) =\left\{f\in C(\mathbb R^n):\ \lim_{x\to\infty} |f(x)| = 0\right\}.$$

I try to prove the Schwarz space is contained in $C_0(R^n)$, but I have no idea how to prove that the Schwarz space satisfies the second condition in $C_0(\mathbb R^n)$. Can someone give me an answer? I will be so grateful.

Appreciate a lot in advance.

Answer 1

Hint:

$$\sup_{x\in\Bbb R^n}\left|x^\alpha \partial_\beta f(x)\right|<\infty\;\;\;\forall\,\alpha\in\Bbb N_0\implies \sup_{x\in\Bbb R^n}\left|xf(x)\right|=M<\infty$$

choosing $\;\beta=0\;$