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}
$\square$
