Have you ever seen the proof that (0,1) (the open interval in R) is uncountable? I think you could do something like that here and get the result you want.
Let be countable sets, and let their Cartesian product x x x ..... be defined to be the set of all sequences where , is an element of . Prove that the Cartesian product is uncountable. Show that the same conclusion holds if each of the sets has at least two elements.
Ok here goes my argument.
Consider that every has at least two elements and consider that the = x x x....... is countable and has elements say .....
now consider the element in the i-th co-ordinate. By the construction of from the very construction of . Hence is an element of the Cartesian product that a contradiction.
Thanks for the help.