The hard part is to show that for any coloring, there's some infinite subset without relying on "well, we can just pick one for each of these infinitely many numbers".

I'm pretty sure that the challenge is to prove that you can for any coloring construct a finitely defined rule for picking the members of the subset which is guaranteed to give you a monochromatic subset.

The question is more about if you can always find such a (finite) rule to partition the set, rather than if you can in a few easily constructed examples.