Why $\frac{1}{2\pi}\int_{-\pi}^{\pi}f(t)g(nt)\ dt$ goes to $\hat{f}(0)\hat{g}(0)$ as $n \to \infty$
Show that $\lim_{n\to \infty}\int_{-\pi}^{\pi}f(x)g(nx)\,dx=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)\,dx\int_{-\pi}^{\pi}g(x)\,dx$