= Solution
Fix $B\Subset\Omega$ and $x_0\in B$. Choose $u_j$ in the Perron family with $u_j(x_0)\uparrow\overline u(x_0)$, replace successive terms by finite maxima using part (i), and take their harmonic lifts on $B$. The lifts remain in the family, are increasing, and are uniformly bounded. Interior estimates and the <Arzela-Ascoli theorem> give a harmonic limit $h$ on $B$ with $h\leq\overline u$ and $h(x_0)=\overline u(x_0)$.
If $h(y)<\overline u(y)$ somewhere in $B$, take another family member larger than $h(y)$ and repeat the maximum-and-lift construction. Its harmonic limit $H$ satisfies $H\geq h$ and $H(x_0)=h(x_0)$. The <strong minimum principle for elliptic operators> forces $H=h$, contradicting the strict inequality at $y$. Thus $h=\overline u$ on $B$. Since $B$ was arbitrary, $\overline u$ is smooth and harmonic in $\Omega$. This is the <Perron method for the Dirichlet problem>.
Back to article page
