(i) The linear algebraic equations $A \mathbf{u}=\mathbf{b}$, where $A$ is symmetric and positive-definite, are solved with the Gauss-Seidel method. Prove that the iteration always converges.

(ii) The Poisson equation $\nabla^{2} u=f$ is given in the bounded, simply connected domain $\Omega \subseteq \mathbb{R}^{2}$, with zero Dirichlet boundary conditions on $\partial \Omega$. It is approximated by the fivepoint formula

$$U_{m-1, n}+U_{m, n-1}+U_{m+1, n}+U_{m, n+1}-4 U_{m, n}=(\Delta x)^{2} f_{m, n}$$

where $U_{m, n} \approx u(m \Delta x, n \Delta x), \quad f_{m, n}=f(m \Delta x, n \Delta x)$, and $(m \Delta x, n \Delta x)$ is in the interior of $\Omega$.

Assume for the sake of simplicity that the intersection of $\partial \Omega$ with the grid consists only of grid points, so that no special arrangements are required near the boundary. Prove that the method can be written in a vector notation, $A \mathbf{u}=\mathbf{b}$ with a negative-definite matrix $A$.