Let \(X_1,X_2,\ldots\) be i.i.d(finite-valued) r.vs with common distribution function \(F\), and let \(F_n\) be the empirical distribution function based on \(X_1,\ldots, X_n\), i.e., \[ F_{n,\omega}(x)=\frac{1}{n}\sum_{k=1}^nI_{\{X_k\le x\}}(\omega), \mbox{ }\mbox{ }\mbox{ }-\infty<x<\infty,\mbox{ } \omega\in \Omega. \] Then, \[ \sup_{-\infty<x<\infty}|F_n(x)-F(x)|\rightarrow 0 \mbox{ }\mbox{ }\mbox{ a.s.,} \] i.e., for almost all \(\omega\), \(F_{n,\omega}\) converges uniformly to \(F\) as \(n\rightarrow \infty\).