I've spent a lot of time thinking about this problem, but I'm running out of time and need a skeleton of the process to solve it if someone wants to tackle it...

Prove:

If such that the set is uncountable.

I've been trying to do this by finding a counterexample to the following sentence

let the set is countable.

If I can find a single counterexample, then the original problem is true. I've looked at unions to see if I can get high enough in the ordinals so that uncountably many things drop to when applying the function, but that would mean is true. Which, according to my professor, it is not.

Thus, I need help.... please.