elementary set theory - Strategies for proving that a set is denumerable?

This concept has been troubling me. For example, I want to prove that

$$\mathbb Q \times \mathbb Q \sim \mathbb N.$$

This is what my professor has told us:

$$\mathbb Q \sim \mathbb N$$

$$\Rightarrow \mathbb Q \times \mathbb Q \sim \mathbb N \times \mathbb N \sim \mathbb N$$
$$\Rightarrow \mathbb Q \times \mathbb Q \sim \mathbb N$$

But this isn't a complete proof because I haven't shown why $\mathbb Q \sim \mathbb N$, and I'm not sure how to do that. I know that if a set is denumerable, that means that it is equinumerous with $\mathbb N$. And I also know that if one is equinumerous with another, that means that there exists a bijection between the two sets. I'm just having trouble putting all of these ideas together into one proof.