Define Schwartz functions class as $$\mathcal{S}(\mathbb{R}^n)=\left\{ \phi \in C^\infty(\mathbb{R}^n) \,\Big|\, \forall \alpha, \beta \in \mathbb{N}_0^n: \; \sup_{x\in\mathbb{R}^n} |x^\alpha D^\beta \phi(x) | <\infty\; \right\}$$
There is another one that typically used is $$\mathcal{S}(\mathbb{R}^n)=\left\{ \phi \in C^\infty(\mathbb{R}^n) \,\Big|\, \forall \alpha, \beta \in \mathbb{N}_0^n: \; \sup_{x\in\mathbb{R}^n} |(1+|x|^2)^{k/2} D^\beta \phi(x) | <\infty\; \right\}$$
They are equivalent since the relation that $(1+|x|^2)^{k/2}\ge x^k$ hence $\mathcal{S}_2\subset \mathcal{S}_1$ conversly we do as follows:
Let's consider the two estimate below $$\sum_{|\alpha| \leq k}\left(\xi^{k}\right)^{2} \leq\left(1+|\xi|^{2}\right)^{k} \leq C(k) \sum_{|\alpha| \leq k}\left(\xi^{k}\right)^{2}$$
And $$D(k)\sum_{|\alpha|\le k}(|\xi|^{\alpha}) \le(\sum_{|\alpha|\le k}(|\xi|^{2\alpha}))^{1/2}\le D'(k)\sum_{|\alpha|\le k}(|\xi|^{\alpha})$$
Where $D,D',C$ are some constant.
Hence we see that $$(1+|x|^2)^{{k}/2}\sim \sum_{|\alpha|\le k }|x|^{\alpha}$$ hence $\mathcal{S_1}\subset \mathcal{S_2}$
Is my proof correct?