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.

throw a ball out into space, and count how long it takes to get back to you in seconds.
i guarantee you, every number from 0 to infinity is equally as unlikely to be the number of seconds it takes for the ball to get back to you.