Find boundary conditions that allow the recursion $f(k)=\frac{1}{2} f(k-1) + \frac{1}{2} f(k+1)$ for $-B
to uniquely determine the function $f(k)=P(S_{\tau}=A|S_o=k)$ where $\tau = \min {\{ n\geq 0: S_n = A \space \text{or}\space S_n = -B}\}$ where $S_n$ is the position of the simple random walker after $n$ steps.
The boundary conditions $f(A) = 1$ and $f(-B)=0$ uniquely determine $f$, but I am not certain as to why these values give the unique solution. I've encountered this in several books and I always took it for granted, but never quite figured out the intuition behind the solution.