I'm trying to prove that if given the innumerable set A and the set B = {x, y}, then A X B is denumerable as well. Proof: All I have so far is that, assuming A is denumerable, we can create the...

I'm trying to prove that if given the innumerable set A and the set B = {x, y}, then A X B is denumerable as well.

Proof:

All I have so far is that, assuming A is denumerable, we can create the sequence A = a₁, a₂, a₃, ... By definition of cross-product, we can then form the sequence A X B = a₁x, a₁y_,
a₂x, a₂y, a₃x, a₃y, ...

It should be clear that this sequence includes every element of AXB as, as such, A X B is denumerable.

I am not entirely sure that this argument is valid.