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.

>>19
There's nothing that needs to converge in the first place.  You just pick a random number in [0,1], and then pick the set that number is in.