Fourier Transform of 1-Lebesgue Space Function is Bounded

Theorem
Let $n \in \N_{>0}$.

Let $\map {L^1} {\R^n}$ be the complex-valued Lebesgue $1$-space with respect to the Lebesgue measure.

Let $f \in \map {L^1} {\R^n}$.

Let $\hat f$ be the Fourier transform of $f$.

Then:
 * $\ds \sup_{\mathbf s \mathop \in \R^n} \cmod {\map {\hat f} {\mathbf s} } \le \norm f_{\map {L^1} {\R^n} }$

Proof
For each $\mathbf s \in \R^n$: