Show set is countable I want to show that the set
$A=\{n^2+m^2:n,m\in \mathbb{N}\}$
is countable.
Is it enough to state that, since $\mathbb{N}$ is closed under multiplication and addition, the set A must be a subset of $\mathbb{N}$; and since any subset of a countable set is countable, A must be countable?

Yes. Unless you define "countable" as "infinite and countable", in which case you need to prove that $A$ is infinite (which shoud be almost trivial).
