I'm trying to show the following but have no idea how to begin. I'm quite new to analysis and multi-index notation.
$$ f \in \mathcal{S} \quad \Longleftrightarrow \quad \forall N \in \mathbb{N}, \alpha \in \mathbb N_0 \text{ a multi-index} \ \exists C_{N,\alpha} > 0: \ \vert \partial^{\alpha} f(x) \vert \leq \frac{C_{N,\alpha}}{(1+|x|)^{N}} $$ where $\mathcal{S}$ denotes the Schwartz space.
For the forward direction I've written down the definition of being in Schwartz space i.e. $$ \sup_{x \in \mathbb{R}^{n}}|x^{\alpha}\partial^{\beta}f(x)| \leq C_{\alpha,\beta} $$ where $ \alpha, \beta \in \mathbb N_0$ are multi-indices