Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space.
Let $Z, X_0, X_1, X_2, \dots \in \mathsf{Random}(\Omega \to \mathbb{R}^D)$ each be a D4383: Random euclidean real number on $P$.

Then the following statements are equivalent

(1)	\begin{equation} X_n \overset{a.s.}{\longrightarrow} Z \quad \text{ as } \quad n \to \infty \end{equation}
(2)	\begin{equation} \forall \, \varepsilon > 0 : \mathbb{P} \left( \bigcup_{n \in \mathbb{N}} \bigcap_{m \in \mathbb{N} : m \geq n} \{ |X_m - Z| \leq \varepsilon \} \right) = 1 \end{equation}
(3)	\begin{equation} \forall \, \varepsilon > 0 : \mathbb{P} \left( \bigcap_{n \in \mathbb{N}} \bigcup_{m \in \mathbb{N} : m \geq n} \{ |X_m - Z| > \varepsilon \} \right) = 0 \end{equation}
(4)	\begin{equation} \forall \, \varepsilon > 0 : \lim_{n \to \infty} \mathbb{P} \left( \bigcap_{m \in \mathbb{N} : m \geq n} \{ |X_m - Z| \leq \varepsilon \} \right) = 1 \end{equation}
(5)	\begin{equation} \forall \, \varepsilon > 0 : \lim_{n \to \infty} \mathbb{P} \left( \bigcup_{m \in \mathbb{N} : m \geq n} \{ |X_m - Z| > \varepsilon \} \right) = 0 \end{equation}
(6)	\begin{equation} \forall \, \varepsilon > 0 : \mathbb{P} \left( \liminf_{n \to \infty} \{ |X_m - Z| \leq \varepsilon \} \right) = 1 \end{equation}
(7)	\begin{equation} \forall \, \varepsilon > 0 : \mathbb{P} \left( \limsup_{n \to \infty} \{ |X_m - Z| > \varepsilon \} \right) = 0 \end{equation}
(8)	\begin{equation} \forall \, \varepsilon > 0 : \lim_{n \to \infty} \mathbb{P} \left( \inf_{m \in \mathbb{N} : m \geq n} \{ |X_m - Z| \leq \varepsilon \} \right) = 1 \end{equation}
(9)	\begin{equation} \forall \, \varepsilon > 0 : \lim_{n \to \infty} \mathbb{P} \left( \sup_{m \in \mathbb{N} : m \geq n} \{ |X_m - Z| > \varepsilon \} \right) = 0 \end{equation}
(10)	\begin{equation} \forall \, \varepsilon > 0 : \mathbb{P}(|X_n - Z| > \varepsilon \text{ infinitely often}) = 0 \end{equation}
(11)	\begin{equation} \forall \, \varepsilon > 0 : \mathbb{P}(|X_n - Z| > \varepsilon \text{ finitely often}) = 1 \end{equation}