Suppose we have to solve $d^2y/dx^2= f(y,x)$ where $f$ is Lipschitz and $y(0) =a, y(1) =b$, using finite difference method, i.e., by discretizing the problem into $y_{i+1} - 2y_i + y_{i-1} = h^2f(y_i,x_i)$, with equal spacing. How do we show it converges, i.e., $\lim_n \max_{i \le n} |y(x_i) - y_i| = 0$? You can assume there is a unique twice differentiable solution, but I would like to know what happens when the solution is not unique as well. This seems quite a bit harder than the initial value problem situation.

I understand this is probably a very basic question in numerical analysis, but I just couldn't find a good reference that covers this general case. There has been a paper on this subject, http://www.jstor.org/stable/2004339?seq=3, but I found their proof for the boundary value problem difficult to follow (note that the proof does not mention a or b at all).