= Solution
Write $A=D+L+U$, with $D$ diagonal, and use only old iterates on the right side. The <Jacobi method> is
$$
\boxed{x_i^{k+1}=\frac1{a_{ii}}\left(b_i-\sum_{j\ne i}a_{ij}x_j^k\right),\qquad
x^{k+1}=-D^{-1}(L+U)x^k+D^{-1}b}.
$$
Strict row diagonal dominance ensures every $a_{ii}\ne0$. The exact solution satisfies the same coordinate relation. For $e^k=x^k-x^*$, subtraction gives
$$
|e_i^{k+1}|\le\frac{\sum_{j\ne i}|a_{ij}|}{|a_{ii}|}\|e^k\|_\infty
\le\gamma\|e^k\|_\infty.
$$
Taking the maximum and iterating proves
$$
\boxed{\|x^k-x^*\|_\infty\le\gamma^k\|x^0-x^*\|_\infty\longrightarrow0}.
$$
This direct contraction proof requires no eigenvector assumption and applies to every strictly diagonally dominant <matrix> with the stated uniform $\gamma<1$.
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