= Solution
Order the interior grid values into a vector. The coefficient matrix has diagonal $4+ch^2$ and off-diagonal entries $-1$ at horizontal and vertical neighbours, with boundary values already zero. Every row is strictly diagonally dominant because $c>0$. The standard convergence theorem for <strict diagonal dominance> gives \b[convergence of both the <Jacobi method> and the <Gauss-Seidel method>]. The matrix is also symmetric positive definite.
For $D=(4+ch^2)I$, the <weighted Jacobi method> is $u^{(r+1)}=u^{(r)}+\omega D^{-1}(b-Au^{(r)})$, with iteration matrix $H_\omega=I-\omega D^{-1}A$. The sine-product grid <eigenvectors> give
$$
\boxed{\lambda_{k,l}=1-\omega\frac{4\{\sin^2(\pi kh/2)+\sin^2(\pi lh/2)\}+ch^2}{4+ch^2},\qquad1\leq k,l\leq m.}
$$
These <weighted Jacobi eigenvalues for a square Dirichlet grid> determine convergence. All <eigenvalues> of $A$ are positive. Thus for fixed $h$ convergence requires $0<\omega<2(4+ch^2)/a_{\max}$, where $a_{\max}=8\cos^2(\pi h/2)+ch^2$. As $h\downarrow0$, this upper bound tends to $1$ from above. Any fixed $\omega>1$ therefore fails on sufficiently fine grids; $\omega\leq0$ also fails. Consequently \b[$0<\omega\leq1$ is necessary for all mesh sizes], and it is sufficient, since $a_{\max}<2(4+ch^2)$ for every finite grid.
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