[Math] Prove that the set of integer coefficients polynomials is countable

Question

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

Answer 1

Hints:

1) Prove that for each $n\ge$ 1 the set $\mathbb Z^n$ is countable. This can be done by induction.

2) Prove (or be aware of the fact) that a countable union of countable sets is countable.

Now, write the set of all polynomials with integer coefficients as a countable union $\bigcup_n P_n$, where $P_n$ is the set of all polynomials with integer coefficients and of degree smaller than $n$.

Prove that each $P_n$ is countable by establishing a bijection between $P_n$ and $\mathbb Z^n$.