I have a feeling that this question is actually rather easy, and that I have all necessary "building blocks". However, I got tremendously confused while actually trying to make the proof, and I'd like some assistance in where to begin and what to use next, etc.
I'm trying to proof this rigorously, with Lebesgue-integration. Since I know that $$\lim_{a \to +\infty}\int_{-a}^{a}\frac{sin(y)}{y}dy = \pi,$$ my plan was to use the dominated convergence theorem, but I'm stuck on the exact way to do that properly in this case. I've made a lot of similar proofs in the past, but I seem to have forgotten the technique. A push in the right direction would be highly appreciated.