1. Imagine a positive point x not equal to zero. 2. Consider a randomly chosen point y with distance to zero less than x. 3. Let y=x. Repeat #2. 4. Is the sum of the y-values finite as y approaches zero?
For step three, it's supposed to be the other way around, right? x is supposed to equal y? Otherwise there's no reason for y to approach zero (or any other number). I don't know if it always converges, but on average it converges to x (by "average" I mean that for any given random y value, the average of all choices is x/2, so y, on average, equals x/2).
Hi, I understand this as follows: denote by ran(x) a random number between 0 and x. Let x_{1}=ran(1), and let x_{i}=ran(x_{i-1}) for x>1. Let S be the sum Ʃx_{i}. As noted above, the expected value of S is 1 (does require a very minor argument). The chance of the series not converging is 0. For example the chance of S>N must be less than 1/N, for the average sum to be 1, so the chance of divergence is less than 1/N for any positive N. (A small simulation shows that the chance of the sum exceeding 7 is about 1 in 10 million)