Bandwidth of the product of two functions?

Question

Let $f(x)$ and $g(x)$ be two functions with finite bandwidth, i.e., the support of their Fourier transform has length $a$ and $b$, respectively. How can we show that the bandwidth of $f(x)g(x)$ is upper bounded by $a+b$? It's not direct to see this result by simply doing the Fourier transform $\frac{1}{2\pi}\int_{-\infty}^\infty f(x)g(x)e^{-iwx}dx$.