This is the third part of a question where the first two parts are to prove that (i) $\mathbb{Z}^n$ is denumerable (ii) $A_n = \{\sum_{k=0}^n x^n : a_k \in \mathbb{Z}\}$ is denumerable. I have these two parts done, the first using induction and the second by simply finding a bijection $f:\mathbb{Z}^{n+1} \rightarrow A^n$ and combining this with the result of (i).
The third part then says to prove that $\mathbb{Z}[X]$ is denumerable. It seems to me that this follows directly from the two previous parts. It seems that part (i) should extend to an infinite tuple and then this could be easily mapped to $\mathbb{Z}[X]$. I am doubtful of this approach because it seems too simple and I know that intuition often doesn't hold when extending to infinity so I am wondering if I am overlooking some problem with this approach?
Hint: let $A_n = \{\sum_{k=0}^n a_kX^k : a_k \in \mathbb{Z}\}$, i.e., the set of all polynomials with integer coefficients of degree at most $n$. First prove $A_n$ is countable. Then prove that $\Bbb{Z}[X] = \bigcup_nA_n$ is countable.