Putnam 1967 B3

If $f$ and $g$ are continuous and periodic functions with period $1$ on the real line, then $$\lim_{n\to \infty} \int_{0}^{1} f(x)g (nx) dx =\left( \int_{0}^{1} f(x) dx\right)\left( \int_{0}^{1} g(x) dx\right).$$