You should be able to solve this.

Create a method to choose a random integer between 0 and infinity such that no integer is more likely to be chosen than any other.

Yes it is possible, I don't care what your probability book says (Notice I didn't even use the word "probability").  You may assume the axiom of choice.

>>21
That part obviously converges, but that's just an intermediate result. There's no convergence after doing the mapping.

>>22
Right, so no convergence. It's all or nothing.

>>16
Assuming that "countable union of disjoint subsets" refers to the method on the everything2 page, you can't say that they have the same size and shape after you've cut them to the [0, 1] range.

I wondered if you could choose the elements in X so as to all have a difference of more than 1, but then you'd have covered a real interval with no more than one element from each of an allegedly countable amount of sets, which can't be good.

Meh, too hot right now, will think more about it later.