Is this proof ok?

If I assume that the set of two-touples is countable (which I think follows directly from the proof the the countable union of countable sets is countable).

Can we make the natural bijection s.t.

. Then we can see that the positive rationals are countable.

Making the similar bijection s.t.

. Then we see that the negative rationals are countable.

If we the take , then as the countable union of countable sets is countable we have that $\mathbb{Q}$ is countable.

Note: I make the bijection to the set of two-touples as it is countable, so it has a bijection to the natural numbers and the composition of bijhections is a bijection.

Thanks for any help