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.

>>12
>F has the same cardinality as the reals
I'm not sure about that, but I'll go with it for now.

So let me see if I get this.  Of the uncountably infinite set of functions from [0,\infty) \rightarrow [0,1], you're selecting a sequence f_i := f(i) (how? just any sequence?), and setting g(x) := \sum_{i=1}^{\infty} f_i(x).  Then g is a uniform limit of continuous functions, and so is continuous, and the output is the next integer above the smallest local maximum.

How do you know g has a local max at all?  Also, where's the "random" part of it?  Are you selecting the f's at random?  How are you doing it?