Countable Product of Separable Spaces is Separable

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 separable.

Then $\left({X, \vartheta}\right)$ is also separable.