Could someone verify my proofs?
Proposition: the set of all finite subsets of $\mathbb{N}$ is countable
Proof 1:
Define a set $ X=\{A\subseteq\mathbb{N}\mid \text{$A$ is finite} \}$.
We can have a function $g_{n}: \mathbb{N} \rightarrow A_{n} $ for each subset such that that function is surjective (by the fundamental theorem of arithmetic). Hence each subset $A_{n}$ is countable.
By the "Union of countable sets is countable" theorem, X is countable.
Q. E. D.
Proof 2: Suppose we have an ordered list of prime numbers $p_{1}, p_{2}, ..., p_{n}$. Define a function $g: X \rightarrow \mathbb{N}$ such that $ g(A) = (p_{1})^{a_{1}}*(p_{2})^{a_{2}}*...*(p_{k})^{a_{k}} $ with $k=$ number of elements of each subset and $ a_{1}, a_{2}, ..., a{k} $ the ordered elements of them. By the fundamental theorem of arithmetic, that function is injective, hence $X$ is countable.
Q. E. D.