We agree! 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 sai

We agree! 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.