Proposition i find in a course : "If $f:\mathbb{R}\rightarrow\mathbb{R}$ is integrable and it's fourier transform is integrable too , then $f$ and $\hat{f}$ are $L^1$ ".
I can't really figure out why it's so trivial. I can't really manage to find out why convergence of $f$ would implie absolute convergence on $\mathbb{R}$..