If $A$ is separable, then there exists a countable, dense subset of $A$ like $\{p^{(i)} \in A:i \in \mathbb{N} \}$. Similarly for separable $B$ as $\{q^{(i)} \in B : i \in \mathbb{N} \}$.
I think the Cartesian product $A \times B$ is separable as well. How do I write the countable, dense subset of $A \times B$?
$\{(\lambda^{(i)}, \mu^{(i)} ) \in A \times B : i \in \mathbb{N} \}$
or
$\{(a^{(i)}, b^{(j)} ) \in A \times B : i \in \mathbb{N}, j \in \mathbb{N} \}$
Are both valid ways of proceeding forward? This is just part of some proof I am working on.
EDIT: I changed the letters to make the point that all the sets are independently formulated.
The first one is wrong. For example, if $A=B$ and $p_i=q_i$ then your set is dense only in the diagonal of $A \times B$ (i.e. it is dense in $\{(x,x): x \in A\}$).