Second-countability is countable product-closed

Statement
A product of countably many second-countable spaces, given the product topology, is also a second-countable space.

Related facts
The result is not true if we give the box topology to the product space.

Proof
Given: A countable family of spaces $$X_n, n \in \mathbb{N}$$, with each $$X_n$$ having a countable basis $$B_{n,m}$$. $$X$$ is the product of the $$X_n$$s, given the product topology

To prove: $$X$$ is a second-countable space