= Solution
The <Riemann-Lebesgue lemma> states that the <Fourier transform> of an $L^1$ function is continuous and tends to zero at infinity.
For a <Schwartz space>[Schwartz function] $f$, differentiation under the integral and <integration by parts> give
$$
ξ^α D^β \widehat f(ξ)=C_{α,β}\widehat{D^α(x^βf)}(ξ).
$$
The <Riemann-Lebesgue lemma> bounds the right side by an $L^1$ seminorm of $D^α(x^βf)$. Every Schwartz seminorm of $\widehat f$ is therefore bounded by finitely many Schwartz seminorms of $f$. Hence the Fourier transform maps $\mathcal S(\mathbb R^n)$ continuously into itself.
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