> The second problem is, what does it mean to choose a real at random ?... So effectively it is impossible to pick a random real number

Yes, it's established there isn't "uniform distribution over all real numbers" without violating axiom of probability. You're 100% correct on this.

But it doesn't make Colin's solution wrong, because e > 0 for any* well-defined distribution.

> Which is a different problem than stated originally

There are two ways to inteprete the original problem:

A. The numbers are truly randomly picked over all real numbers.

B. The numbers are picked from a well-defined distribution which is unknown to the player.

Since A. is invalid mathematically speaking (without changing the commonly accepted definition of probability), it's reasonable to only consider B., in which case, Colin's solution is correct.

I made a more intuitive explantion on why a strategy better than coin toss exists here: https://news.ycombinator.com/item?id=42372972

*: More strictly, any distribution that guarantees the probability that the two numbers in envelope are the same = 0.