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.

w is the first infinite ordinal (The natural numbers). c is the first ordinal with the same cardinality as the real numbers. f is a function defined from c to F, so f(i) is a continuous function. F has the same cardinality as the reals, there is a easy injection from F to Q^Q, where Q is the set of rational numbers.