I want to show that $e^{-a\sqrt{1+\|x\|^2}}$ is in $\mathcal{S}(\mathbb{R}^d)$. ($a>0$)
Please tell me proof.
Where,
$$\|x\|^2 = \left(\sum_{j=1}^{d} |x_j|^2\right)$$
$$ f(x) \in \mathcal{S} \overset {\mathrm{def}} {\Leftrightarrow} \displaystyle \sup_{x \in \mathbb{R}^d} |x^\alpha\partial^\beta_x f(x)| < \infty $$
$\alpha,\beta$ is multi index notations.
Hint: Show by induction that every derivative of $f$ is a finite sum of functions of the form
$$p(x)(1+|x|^2)^rf(x).$$
Here $p$ is a polynomial and $r\in \mathbb R.$