We have a simple random walk S on a finite set of $\mathbb{N}: {0,...,N}$.
Let $m_x$ be the expected number of steps, starting at $x$, required to reach $0$ or $N$ for the first time. Show that $m_x$ satisfies the conditions:
(a) $m_0 = 0$
(b) $m_N = 0$
(c) $m_x = \frac{1}{2}m_{x-1}+\frac{1}{2}m_{x+1}+1$.
For (a) and (b) my reasoning goes as follows. Substitute either $0$ or $N$ for $x$ in $m_x$. Then we are looking for the expected numbers of steps before reaching either $0$ or $N$ starting in either $0$ or $N$. This is $0$ since we are already there.
How do we show that the difference equation applies though?