In the unit square the Poisson equation $\nabla^{2} u=f$, with zero Dirichlet boundary conditions, is being solved by the five-point formula using a square grid of mesh size $h=1 /(M+1)$,

$$u_{i, j-1}+u_{i, j+1}+u_{i-1, j}+u_{i+1, j}-4 u_{i, j}=h^{2} f_{i, j} .$$

Let $u(x, y)$ be the exact solution, and let $e_{i, j}=u_{i, j}-u(i h, j h)$ be the error of the five-point formula at the $(i, j)$ th grid point. Justifying each step, prove that

$$\left[\sum_{i, j=1}^{M}\left|e_{i, j}\right|^{2}\right]^{1 / 2} \leqslant c h, \quad h \rightarrow 0$$

where $c$ is some constant.