I have been trying to prove this. I came up with two cases. One case where the inifnite set is countable and one where it isn't. For the one that isn't. I said that an infinite set must contain a countably infinite set and that the difference of the original set and this new countably infinite set must be still be uncountable. Therefore, by induction, we can find a pair of disjoint infinite sets. However, I am having trouble proving the union is countably infinite.
For the countable side, I know it should be trivial but my mind is still coming up blank.
Any help would be appreciated thank you!
