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
Where does Axiom of Choice actually come in? AC implies "there exists" such an infinite sequence, but you're already defining one hence don't need it.