I have to show if function is in schwarz space $S(\mathbb R)$, then it is also in $C_0(\mathbb R )$(continuous functions dying at infiinity).
$f\in S(\mathbb R) $means $f$ is infinitely many times differentiable and for any $n,m \in \mathbb N_0$, sup$\mid x^mf^n(x) \mid$ is finite over $\mathbb R$. Now clearly $f$ from schwarz space is continuous but how can I show that $\forall \epsilon>0$, there exist compact set $K_\epsilon$, such that outside $K_\epsilon$, $\mid f(x) \mid< \epsilon$, which is condition for $C_0(\mathbb R )$. I am not coming with any idea about the construction of $K_\epsilon$. Any hint.