Proof verification : $\mathbb{Q}$ is countable

Question

To show that : $\mathbb{Q}$ is countable

Set $A$ is said to be countable if there exists a bijection from $A$ to $\mathbb{N}$. Every countable set is infinite.

Further I have proved the following results:

1. Set $A$ is finite or countable $\iff$ $\exists f:N\to A $ a surjection $\iff \exists g:A\to N$ an injection.

2. Let $f:S\to T$ an injection where $T$ is countable and $S$ is infinite, then $S$ has to be countable. This result follows directly from 1.

$\mathbb{Q}= \{p/q \; | \; p\in Z,\; q\in N, (p,q)=1\}$

define. $f:\mathbb{Q}\to \mathbb{Z} \times \mathbb{N}$ as the identity map which is an injection too.

Now, $\mathbb{Q}$ is infinite as $\mathbb{N}\subset \mathbb{Q}$ and $\mathbb{N} $ is countable.

Also $\mathbb{Z}\times\mathbb{N}$ is countable as $\mathbb{Z}$ and $\mathbb{N}$ are countable and using the fact finite product of countable is countable.

Hence by 2. above $\mathbb{Q}$ is countable.

Is this proof okay?

Answer 1

Alternate proof.
As |Q| <= |Z×Z| and the latter is countable, Q is countable.
Since Q is infinite, it is countablely infinite, hence equinumerous to N.