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.

Let F be the set of all continuous functions from [0,inf] to [0,1]. Let f be an biyective function from c to F.
Consider g a (continuous) function from [0,inf] to [0,1] defined by:
g(x) = sum from i=i to w of (1/2^i)f(i)(x). Let y = inf { x in R : g(x) is maximum }.

Choose n = min { m in N : m > y}

Is that what you wanted OP?