Countable Product of First-Countable Spaces is First-Countable

Theorem
Let $\left \{{\left({X_\alpha, \vartheta_\alpha}\right)}\right\}$ be a countable set of topological spaces.

Let $\displaystyle \left({X, \vartheta}\right) = \prod \left({X_\alpha, \vartheta_\alpha}\right)$ be the product space of $\left \{{\left({X_\alpha, \vartheta_\alpha}\right)}\right\}$.

Let each of $\left({X_\alpha, \vartheta_\alpha}\right)$ be first-countable.

Then $\left({X, \vartheta}\right)$ is also first-countable.