Let $\lambda > 0$ and let $f \in L^1(\mathbb{R}) \cap \mathcal{C}^1(\mathbb{R})$ such that $\hat{f} \in L^1(\mathbb{R})$ and such that $\hat{f}$ is zero outside the segment $[-\lambda, \lambda]$. Deduce that, if $f \in L^\infty(\mathbb{R})$, there exists a constant $C \in \mathbb{R}_+^*$, independent of $\lambda$ and of $f$, such that
$$\left\|f'\right\|_\infty \leqslant C\lambda \|f\|_\infty$$