Prove that the set of integer coefficients polynomials is countable How to prove that the set of integer coefficient polynomials is countable?

$1$. Prove that the set $S_n$ consisting of polynomials of degree $n$ with integer coefficients is countable. Show this by constructing a bijection between $S_n$ and $\underbrace{\mathbb{Z} \times \mathbb{Z} \times \cdots \times \mathbb{Z}}_{n \text{ times}}$.
$2$. Now the set you are interested in is $S = \bigcup_{n \in \mathbb{N}} S_n$. Show that $S$ is countable by proving that a countable union of countable sets is again countable.
