Let $X$ and $Y$ be random variables on the same probability space. The $\infty$-Wasserstein distance between $X$ and $Y$ is defined as

$$d_{\infty}(X, Y) = \inf \|X_1 - Y_1\|_{L_{\infty}},$$

where the infimum is over all random variables $X_1$ and $Y_1$ with same distribution as $X$ and $Y$, respectively.

Let $F_X$ and $F_Y$ be the cumulative distributions of $X$ and $Y$. I need help proving that

$$d_{\infty}(X, Y) = \inf\{\epsilon > 0 : F_X(t - \epsilon) \leq F_Y(t) \leq F_X(t + \epsilon) \mbox{ for all } t \in \mathbb{R}\}$$

Can someone help me?

**Reference:** Aubrun and Szarek, *Alice and Bob Meet Banach: The Interface of Asymptotic Geometric Analysis and Quantum Information Theory*, pg 161.