Since $E_1, \ldots, E_N$ are each finite, then
\begin{equation}
|E_1| = K_1,
\quad \ldots, \quad
|E_N| = K_N
\end{equation}
for some $K_1, \ldots, K_N \in \mathbb{N}$. Result
R1832: Cardinality of a finite cartesian product of finite sets shows that
\begin{equation}
\left| \prod_{n = 1}^N E_n \right|
= \prod_{n = 1}^N |E_n|
= \prod_{n = 1}^N K_n
< \infty
\end{equation}
$\square$