Prove smooth functions vanishing at infinity belong to Schwartz space

Question

I believe the definition of $C^{\infty}_0(\mathbb R^n)$ is $$\{f(x):\forall\alpha\in\mathbb N^n,D^{\alpha}f(x)\in C_0 \cap C^{\infty}\}$$ note that $\alpha$ is a multi-index with nonnegative integer coordinates. I want to prove these functions are in Schwartz space $\mathcal S$ $$\mathcal S:=\{f(x)\in C^\infty:\sup_{x\in\mathbb R^n}|x^{\alpha}D^{\beta}f(x)|\lt\infty \}$$ It is clear from the definition that $D^\beta f(x)\in C_0^\infty$ if $f(x)\in C_0^\infty$ thus $\sup_{x\in\mathbb R^n}|D^\beta f(x)|<\infty$. How do I show that for any $\alpha\in \mathbb N^n$, $\sup_{x\in \mathbb R^n} |x^\alpha f(x)|\lt \infty$? In the case I wrongly stated some definitions, do correct me.