= Quantile proof of the Glivenko-Cantelli theorem
{c}
For a <cumulative distribution function> $G$, let $Q(u)=\inf\{x:G(x)\geq u\}$, $0<u<1$. Right continuity gives $Q(u)\leq x$ exactly when $u\leq G(x)$, including at atoms. Thus applying $Q$ to independent uniform variables $U_i$ constructs an independent sample with distribution function $G$. Its <empirical distribution function> is $G_n(x)=F_n(G(x))$, where $F_n$ is the uniform-sample empirical distribution function. Consequently $\sup_x|G_n(x)-G(x)|\leq\sup_{t\in[0,1]}|F_n(t)-t|$. For the uniform sample, monotonicity and a grid of mesh $1/m$ bound the last supremum by the largest error at grid points plus $1/m$. The <strong law of large numbers> on the countable union of all such grids proves the almost-sure limit. This construction proves the theorem for distributions with atoms as well as continuous distributions.
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