Countable Product of First-Countable Spaces is First-Countable

Theorem
Let $I$ be an indexing set with countable cardinality.

Let $\left\langle{\left({S_\alpha, \tau_\alpha}\right)}\right \rangle_{\alpha \mathop \in I}$ be a family of topological spaces indexed by $I$.

Let $\displaystyle \left({S, \tau}\right) = \prod_{\alpha \mathop \in I} \left({S_\alpha, \tau_\alpha}\right)$ be the product space of $\left\langle{\left({S_\alpha, \tau_\alpha}\right)}\right \rangle_{\alpha \mathop \in I}$.

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

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