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
You flip a coin an infinite amount of times to decide a, for the sake of simplicity, binary decimal expansion.

So You have numbers X_i

X_1 = 0.a_1
X_2 = 0.a_1a_2
X_3 = 0.a_1a_2a_3

etc

We define the number we pick to be X = Lim X_i as i-> inf

Now obviously X_i is a cauchy sequence and |X_i - X_j| is at most 1/2^i for all j > i

So the sequence converges. This is all fine with the axiom of choice.


The other way though, if we pick digits going to the right, you can easily see that the sequence you get is not cauchy and doesn't converges.


As a very simple example what value do you give to the number if all your coin flips are heads?