Let be the fortune of a gambler at time t. It is assumed that is a discrete time Markov process with state space {0,1,2,...N} where N is a positive integer. For

States 0 and N are absorbing so that

You may assume that is a martingale with respect to itself.

(i) Use the Martingale Convergence Theorem to show that with probability 1 the process is eventually absorbed at 0 or N.

I cannot see how the MCT can show that the process is eventually absorbed at 0 or N as it only shows that for some