Yeah I wrote it last night and wasn't paying attention to what I was writing. The function should instead be a function from N X N [tex]\rightarrow[/tex] A X B. Other than that little error the principle is the same. [tex]\zeta[/tex]( n[tex]_{1}[/tex] , n[tex]_{2}[/tex] ) = ( [tex]\alpha[/tex]( n[tex]_{1}[/tex] ) , [tex]\varphi[/tex]( n[tex]_{2}[/tex] ) ). This function is a bijection and N X N is countable so any cartesian product of two countably infinite sets is countable.Good point, in fact, it is very badly non surjective!
For example, if [itex]\alpha(2)= p[/itex] and [itex]\beta(2)= q[/itex], then [itex]\zeta(2)= (p, q)[/itex] but there is NO other [itex]\zeta(n)[/itex] giving a pair having p as its first member nor is there any other n so that [itex]\zeta(n)[/itex] is pair having q as its second member.