Let $\displaystyle X_t$ be the fortune of a gambler at time t. It is assumed that $\displaystyle (X_t : t=0,1,2...)$ is a discrete time Markov process with state space {0,1,2,...N} where N is a positive integer. For $\displaystyle 1 \leq i \leq N-1$

$\displaystyle \mathbb{P}(X_{t+1} = i-1 | X_t = i)=\frac{1}{2}= \mathbb{P}(X_{t+1}=i+1 | X_t=i)$

States 0 and N are absorbing so that

$\displaystyle \mathbb{P}(X_{t+1} = 0 | X_t = 0)=1=\mathbb{P}(X_{t+1}=N | X_t=N)$

You may assume that $\displaystyle (X_t)$ 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 $\displaystyle X_t \rightarrow X$ for some $\displaystyle X$