Given a continuous-time, discrete space approximation of $u_t = u_{xx}$ with homogeneous Dirichlet boundary condition. The discrete ODE version for the nodal values is: $u'_j = \frac{u_{j-1}+u_{j+1} - 2u_j}{(\delta x)^2}$ for $j = 1,\dots, N-1$ over the domain $(0, N\delta x)$ where $\delta x = 1/N$. Note that $u_j (t) = u(j\delta x, t)$, and $u_0(t) = u_N(t) = 0$.
Question: Prove the discrete maximum principle, which is: If $w_j(t)$ for $j = 1,\dots, N-1$ solves the ODE above and $w_0(t) = w_N(t) = 0$, then max$_{j, t} w_j(t)$ and min$_{j, t} w_j(t)$ are either achieved at $t= 0$ or $j = 0$ or $j=N$.
My progress: I have tried using contradiction method to mimic the proof in the continuous case for the PDE: $u_t - \Delta u = 0$, but failed to reach anywhere. Can anyone please help me with this problem?