\begin{align} \partial_t u(x,t) & = \kappa \partial_{xx}u(x,t), & -1 < x < 1, & \quad t>0 \quad \kappa > 0 \nonumber\\ u(-1,t) & = g_1(t) & t>0 & \\ u(1,t) & = g_2(t) & t> 0 & \nonumber \\ u(x,0) & = n(x) & -1 \leq x \leq 1 \nonumber \end{align}
In order to solve the problem (1D heat equation) above I use the FTCS-scheme. $\theta$ is chosen such that $\theta = 0$. I have proofed that the scheme is consistent of order $\mathcal{O}(k+h^2)$ and furthermore proofed that the scheme is stable for the choice $$ \mu = \frac{\kappa k}{h^2} \leq \frac{1}{2}$$ However showing that the scheme is convergent, as well as showing that the scheme satisfies the discrete maximum principle for a special choice of $\mu$, is proofing difficult. How would one go about doing it?