Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that

(i)	$E_0, E_1, E_2, \ldots \in \mathcal{F}$ are each an D1716: Event in $P$
(ii)	\begin{equation} \mathbb{P}(E_0) = 1 \end{equation}

Result R4143: Countable union is an upper bound to each set in the union shows that \begin{equation} E_0 \subseteq \bigcup_{n \in \mathbb{N}} E_n \end{equation} Thus, as a consequence of R2090: Isotonicity of probability measure, we have the lower bound \begin{equation} 1 = \mathbb{P}(E_0) \leq \mathbb{P} \left( \bigcup_{n \in \mathbb{N}} E_n \right) \end{equation} Since also $\mathbb{P} ( \bigcup_{n \in \mathbb{N}} E_n ) \leq 1$, then result R1043: Equality from two inequalities for real numbers states that we have \begin{equation} \mathbb{P} \left( \bigcup_{n \in \mathbb{N}} E_n \right) = 1 \end{equation} $\square$