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 $\displaystyle n$ range freely over the naturals.
b) The second part confuses me. The best way to do this is to first prove that if $\displaystyle f:\mathbb{N}\mapsto E$ is surjective, then $\displaystyle E$ is countable and then note that $\displaystyle g:\mathbb{N}^2\mapsto\bigcup_{j=1}^{\infty}A_n$ where $\displaystyle g(m,n)=f_m(n)$ (where $\displaystyle f_m$ is the bijection from $\displaystyle \mathbb{N}$ to $\displaystyle A_m$) is a surjection.