Exercise 3.6.2 of Durrett’s *Probability: Theory and Examples*, 4th edition is

Show that \(\|\mu-\nu\| \le 2\delta\) if and only if there are random variables \(X\) and \(Y\) with distributions \(\mu\) and \(\nu\) so that \(P(X \ne Y) \le \delta\).

The first inequality should read \(\|\mu-\nu\| \le \delta\).