the first one is this :
I am working with Fourier inverse theorem and I have $f(t)=\frac{1}{2 \pi} \int\left(\int f(u) e^{-i \omega u} d u\right) e^{i \omega t} d \omega$ with $f \in L^1(\mathbb{R})$ the book says I can't swap the integral because $f(u) e^{i \omega(t-u)}$ is not in $L^1(\mathbb{R} \times \mathbb{R})$ but why ? mabe because \begin{aligned} & \left|f(u) e^{i \omega(t-u)}\right|=|f(u)| \\ & \iint|f(u)|=\int_{-\infty}^{\infty} c \stackrel{?}{=} \infty \end{aligned}
and the second one is this : How can I find $\sup _{n \in R}\left|\frac{e^{-i n}-1}{n}\right|$? How can I show this is finite ?
I have to do derivatie of complex exponential as usual ? but calculations are very bad