Proposition 5.5.7.label Let $\seqf{X_j}$ be separable topological spaces, then $\prod_{j = 1}^{n} X_{j}$ is also separable.
Proof. Let $\seq{A_j}$ such that $A_{j} \subset X_{j}$ is countable and dense for each $1 \le j \le n$. By Proposition 5.5.6, $\prod_{j = 1}^{n} A_{j}$ is countable and dense in $\prod_{j = 1}^{n} X_{j}$.$\square$