Extending symbol of pseudodifferential operator defined outside of a ball

Question

While reading Wong's book "An introduction to pseudo-differential operators" (3rd edition), I came across the following statement in the exercises :
Let $\sigma \in C^\infty(R^n\times R^n) $ and $m \in R$. Prove that $\sigma\in S^m $ if and only if for all multi indices $\alpha,\beta$, there exists positive constants $C$ and $R$ (depending on $\alpha,\beta$) such that :
$|(D^\alpha_x D^\beta_\xi\sigma)(x,\xi)|\leq C(1+|\xi|)^{m-|\beta|}, \forall |\xi|\geq R$
Here, $S^m$ are symbols satisfying the previous inequality $\forall \xi \in R^n$.
I really don't see how to do this (ie show the inequality is true in general when it's true outside the ball $|\xi|\geq R$), even when $n=1,\alpha=\beta=0$. More precisely, the real issue for me is that $x$ is not necessarily contained in a compact subset(ie that it's not assumed that $\sigma$ vanishes outside of a compact subset in the $x$ variable).
Thank you for any help.