= Solution
The <Glivenko-Cantelli theorem> states that for the empirical distribution function $F_n$ of iid real observations with distribution function $F$,
$$
\sup_{t\in\mathbb R}|F_n(t)-F(t)|\longrightarrow0
$$
almost surely.
Apply part (a) to $\mathcal G=\{\mathbf1_{(-\infty,t]}:t\in\mathbb R\}$. For each $\varepsilon>0$, choose finitely many quantile cutpoints so that the $P$-mass between consecutive cutpoints is at most $\varepsilon$, treating atoms as cutpoints themselves. Indicators at adjacent cutpoints give finite $L^1(P)$ brackets of width at most $\varepsilon$. Using the strong law for the finite bracket endpoints and then intersecting the probability-one events for $\varepsilon=1/m$ gives the almost-sure conclusion.
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