Product space of separable Normed spaces

Question

I want to show that if two normed spaces $X$ and $Y$ are separable then there product space $Z=X \times Y$ is also separable.

Answer 1

Since $X$ is separable there exists $A\subseteq X$ such that $A$ is dense in $X$ and countable.

Since $Y$ is separable there exists $B\subseteq Y$ such that $B$ is dense in $Y$ and countable.

Hint: Prove that $A\times B$ is dense in $X\times Y$ and countable.

Answer 2

Hint 1: $\overline{A \times B}=\overline{A} \times \overline{B}$. A proof of this can be found here.

Hint 2: Countable product of finite sets is countable.