not converge, then and differ. If also the sequence is bounded, then there are some real numbers and such that and the sequence crosses the interval infinitely often. That is, the sequence is eventually less than , and at a later time exceeds , and at an even later time is less than , and so forth ad infinitum. These periods where the sequence starts below and later exceeds are called "upcrossings". Consider a stock market game in which at time , one may buy or sell shares of the stock at price . On the one hand, it can be shown from the definition of a supermartingale that for any there is no strategy which maintains a non-negative amount of stock and has positive expected profit after playing this game for steps. On the other hand, if the prices cross a fixed interval very often, then the following strategy seems to do well: buy the stock when the price drops below , and sell it when the price exceeds . Indeed, if is the number of upcrossings in the sequence by time , then the profit at time is at least : each upcrossing provides at least profit, and if the last action was a "buy", then in the worst case the buying price was and the current price is . But any strategy has expected profit at most , so necessarily By the monotone convergence theorem for expectations, this means that so the expected number of upcrossings in the whole sequence is finite. It follows that the infinite-crossing event for interval occurs with probability . By a union bound over all rational and , with probability , no interval exists which is crossed infinitely often. If for all there are finitely many upcrossings of interval , then the limit inferior and limit superior of the sequence must agree, so the sequence must converge. This shows that the martingale converges with probability . Failure of convergence in mean Under the conditions of the martingale convergence theorem given above, it is not necessarily true that the supermartingale converges in mean (i.e. that ). As an example, let be a random walk with . Let be the first time when , and let be the stochastic process defined by . Then is a stopping time with respect to the martingale , so is also a martingale, referred to as a stopped