Did I prove this sufficiently?
You mean union.
a) Proving by induction implies that the union of a fixed arbitrarily large number of countable sets is countable. It says nothing about letting range freely over the naturals.
b) The second part confuses me. The best way to do this is to first prove that if is surjective, then is countable and then note that where (where is the bijection from to ) is a surjection.