Prove Theorem 3.17. That is, let \(F\) and \(G\) be two distribution functions. Show that

\[\|F-G\|_{\infty}=\sup _{t \in \mathbb{R}}|F(t)-G(t)|\]

is a metric in the space of distribution functions.

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Theorem 3.17. Let F and G be two distribution functions. Then, do (F,G)=||F- G|| = sup |F(t) - G(t)| tR is a metric in the space of distribution functions called the supremum metric.