Just to confirm, I am using 'almost surely' in the technical sense, which means 'with probability 1.'
Consider the following statement:
If you keep flipping a fair coin every day, it is almost sure that after some day you will have gotten a tails.
This is the same 'almost surely' that I am referring to.
The point was that you didn't specify "almost surely" previously, that's why I asked for a proof to understand what did you mean exactly when you said that "everyone goes bankrupt and will never recover" and "or "there is a finite time after which no-one ever passes above $0.0000000000000001".
The mean of a random variable that is close to zero is close to zero, the mean of a random variable that is almost surely close to zero can be anything.