The Poisson equation $\nabla^{2} u=f$ in the unit square $\Omega=[0,1] \times[0,1], u=0$ on $\partial \Omega$, is discretized with the five-point formula

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

where $1 \leqslant i, j \leqslant M, u_{i, j} \approx u(i h, j h)$ and $(i h, j h)$ are grid points.

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

$$\|\mathbf{e}\|=\left[\sum_{i, j=1}^{M}\left|e_{i, j}\right|^{2}\right]^{1 / 2} \leqslant c h \quad \text { for sufficiently small } h>0,$$

where $c$ is some constant independent of $h$.