We want to show if $X, Y$ are random variables defined on a common probability space, with characteristic functions $f, g$ respectively, then the following inequality is valid:
$$\sup |f(x)-g(x)| \le 2P(X\neq Y).$$
This is from an old qualifying exam and I cannot solve it. I tried to analyze the difference in probability measures, but to no avail. Any help would be greatly appreciated.