Let $X_1, X_2, X_3, \ldots \in \text{Random}(\Omega \to \mathbb{R})$ each be a D3161: Random real number such that

(i)	$X_1, X_2, X_3 \ldots$ is an D3358: I.I.D. random collection

Then \begin{equation} \lim_{N \to \infty} \sup_{x \in \mathbb{R}} \left| \frac{1}{N} \sum_{n = 1}^N I_{\{ X_n \leq x \}} - \mathbb{P}(X_1 \leq x) \right| \overset{a.s.}{=} 0 \end{equation}

Let $X_1, X_2, X_3, \ldots \in \text{Random}(\Omega \to \mathbb{R})$ each be a D3161: Random real number such that

(i)	$X_1, X_2, X_3 \ldots$ is an D3358: I.I.D. random collection
(ii)	$F : \mathbb{R} \to [0, 1]$ is a D205: Probability distribution function for $X_1$

Then \begin{equation} \lim_{N \to \infty} \sup_{x \in \mathbb{R}} \left| \frac{1}{N} \sum_{n = 1}^N I_{\{ X_n \leq x \}} -F(x) \right| \overset{a.s.}{=} 0 \end{equation}

Let $X_1, X_2, X_3, \ldots \in \text{Random}(\Omega \to \mathbb{R})$ each be a D3161: Random real number such that

(i)	$X_1, X_2, X_3 \ldots$ is an D3358: I.I.D. random collection
(ii)	$F : \mathbb{R} \to [0, 1]$ is a D205: Probability distribution function for $X_1$

Then \begin{equation} \mathbb{P} \left( \lim_{N \to \infty} \sup_{x \in \mathbb{R}} \left| \frac{1}{N} \sum_{n = 1}^N I_{\{ X_n \leq x \}} -F(x) \right| = 0 \right) = 1 \end{equation}