The proof seems to concentrate on the marginal distribution as n goes to infinity. But the simulation hints at something more interesting: each sample of the ra

The proof seems to concentrate on the marginal distribution as n goes to infinity. But the simulation hints at something more interesting: each sample of the random process seems to converge to a value, where the value itself is U(0,1). Is it true that a sample of the random process is convergent with probability 1?