Countable Product of First-Countable Spaces is First-Countable

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

Let $\family{\struct{S_\alpha, \tau_\alpha}}_{\alpha \mathop \in I}$ be a family of topological spaces indexed by $I$.

Let $\displaystyle \struct{S, \tau} = \prod_{\alpha \mathop \in I} \struct{S_\alpha, \tau_\alpha}$ be the product space of $\family{\struct{S_\alpha, \tau_\alpha}}_{\alpha \mathop \in I}$.

Let each of $\struct{S_\alpha, \tau_\alpha}$ be first-countable.

Then $\struct{S, \tau}$ is also first-countable.