Let $f \in C^\infty(\mathbb R)$. Define the semi-norm $ \|f\|_{a,b}=\sup_{x \in \mathbb R} |x^af^{(b)}(x)| $ where $a,b \in \mathbb Z_+$, and $f^{(b)}$ is the $b$-th derivative of $f$.
Show $\|\cdot\|_{a,b}$ is a norm on the Schwartz space $S(\mathbb R)$.
I don't see how to prove this direction of the nonnegativity requirement for the norm $ \|f\|_{a,b} =0 \text{ implies } f=0. $
If one of the derivative is zero, how can I infer that the original function is also zero?