Definition (Schwartz Space) We say $f \in S(\mathbb{R}) \Leftrightarrow \sup_{x \in \mathbb{R}} |x^i D^j f(x)|< \infty \Leftrightarrow \sup_{x \in \mathbb{R}}|(1+|x|)^n D^k f(x)| < \infty $
for all $i,j,k,n \in \mathbb{N_0}$.
Definition (Fourier Trnsformation): Let $f \in S(\mathbb{R})$ be a Schwartz function then, $\widehat{f}(\xi):=\int_{\mathbb{R}}f(x) e^{-2 \pi i x \xi}dx$ is called the Fourier transform of $f$.
I am struggling with the following: Let $f \in S(\mathbb{R})$ then the Fourier transform exists.
$\int_{\mathbb{R}}f(x)dx\leq \int_{\mathbb{R}}|f(x)|dx=\int_{\mathbb{R}}(1+x^2)|f(x)|\frac{1}{1+x^2}dx=\int_{\mathbb{R}}(1+x^2) |f(x)|dx \int_{\mathbb{R}}\frac{1}{1+x^2}dx=(\int_{\mathbb{R}}(1+x^2) |f(x)|dx) \pi$
To be able to conclude that the Fourier Trnsformation does exist on $S(\mathbb{R})$ As far as I know I somehow have to show that $\int_{\mathbb{R}}(1+x^2) |f(x)|dx$ is bounded, then the prove would be finished.
I don't know how to continue from here on. Could someone please show me (in detail) how to continue from here on. (If my attempt is complety wrong, then an attempt that is working.)