Is set of points (p,q) countable?

[seyma]

I want to show that the set of points (p,q) on the plane with rational coordinates p and q is countable. I proved set of rational numbers is countable by drawing table and I find (http://web01.shu.edu/projects/reals/infinity/proofs/combctbl.html ) Combining Countable Sets. However, I cannot put these together in table.
Could it be proven by matematically instead of table drawing?

 

[CRGreathouse]

If you know how to prove that there are only countably many rationals, the proof is exactly the same.

 

[HallsofIvy]

For that matter, since you have proven that there are only countably many rationals, any subset of the rationals is either finite or countable.