Question: Prove the deterministic lemma we used in the proof of the GlivenkoCantelli Theorem.

Lemma. If \(F_{1}, F_{2}, \ldots, F\) are distribution functions and

1. \(F_{n}(x) \rightarrow F(x)\) for each rational \(x\)
2. \(F_{n}(x) \rightarrow F(x)\) and \(F_{n}(x-) \rightarrow F(x-)\) for each atom \(x\) of \(F\) then \(\sup _{x}\left|F_{n}(x)-F(x)\right| \rightarrow 0\)

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