= Solution
Fix $B=B(x_0,r)\Subset\Omega$. Solvability of the <Dirichlet problem> on a ball gives a unique <harmonic function> $h\in C^2(B)\cap C^0(\overline B)$ with $h=u$ on $\partial B$. Both $u$ and $h$ have the <mean value property for harmonic functions>, so $v=u-h$ has it as well and vanishes on $\partial B$.
If $v$ were not zero, compactness of $\overline B$ would give either a positive maximum or a negative minimum in the interior. Part 1(i) would make $v$ constant, contradicting its zero boundary values. Hence $u=h$ on $B$. Every point lies in such a ball, so
$$
\boxed{\Delta u=0\text{ throughout }\Omega.}
$$
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