Proving function is Schwartz

Question

I want to prove that $f(t)=e^{-t^2C\pi}$ is Schwartz. I tried computing derivatives and showing that for all $n,k\in \mathbb{N}_0$ $$\lim_{t\to \infty}t^{k}f^{(n)}(t)=0$$ but it gets messy pretty fast. Is there another way?

Answer 1

It is easy to see that $f^{(n)}(t)=P_n(t)\,e^{-C\pi t^2}$, where $P_n$ is a polynomial of degree $n$. Now you have to prove that $\lim_{t\to\infty}t^ke^{-C\pi t^2}=0$ for all $k\in\mathbb N$.