= Solution
For $\omega=1$, the <Jacobi method> iteration matrix $T=I-D^{-1}A$ has zero diagonal, nonnegative axial entries $1/5$, and diagonal-neighbour entries $1/20$. Its row sums are at most one; they are strictly below one at grid points next to the Dirichlet boundary. Its graph is connected by the positive axial entries.
To prove the strict <spectral radius> bound, let $Tz=\lambda z$ with $z\ne0$ and choose a component of maximum magnitude $M>0$. Then
$$
|\lambda|M\leq\sum_wT_{vw}|z_w|\leq M\sum_wT_{vw}\leq M,
$$
so $|\lambda|\leq1$. If equality held, every positive-weight neighbour of a maximal component would also have modulus $M$, and that row would have sum one. Propagating along axial edges reaches a boundary-adjacent row, whose sum is strictly less than one, a contradiction. This also covers a one-point interior grid, for which $T=0$. Hence
$$
\boxed{\rho(T)<1.}
$$
The standard convergence theorem for a stationary linear iteration says it converges from every initial vector exactly when its iteration matrix has spectral radius below one. Since $A$ is invertible by (i), the error satisfies $e^{(r)}=T^re^{(0)}\to0$ and
$$
\boxed{\omega=1\text{ converges to the unique solution of }Au=b.}
$$
Weak row diagonal dominance alone would not prove convergence; the connection to a strictly deficient boundary row supplies the required strictness.
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