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Fix a compact $K\subset \Omega$ and choose $\delta >0$ such that a closed $3\delta$-neighbourhood of $K$ is compactly contained in $\Omega$. The hypothesis gives a common bound $M$ on that neighbourhood. Choose a smooth radial mollifier ${\rho }_{\delta }$ supported in $B_{\delta }(0)$ with integral one. The mean value property for harmonic functions, integrated against its radial weights, givesin a neighbourhood of $K$. Differentiating this fixed-radius convolution gives the uniform interior estimateThus the sequence is equicontinuous there; taking its pointwise limit gives the same local modulus of continuity for $u$. A finite small-ball cover of $K$ now proves uniform convergence directly: choose the cover so both $u_j$ and $u$ vary by less than $\epsilon$ inside each ball, then use convergence at its finitely many centres to make $|u_j-u|<\epsilon$ there. The triangle inequality gives ${\sup }_K|u_j-u|<3\epsilon$ for large $j$.

Finally, pass to the limit in the reproducing convolution identity on smaller compact sets. It yields $u=u\ast {\rho }_{\delta }$, so $u$ is smooth locally. Also, for each compactly supported smooth test function $\varphi$, uniform convergence on its support givesHence its smooth Laplacian is zero. We have proved $u_j\to u$ uniformly on every compact subset and $u$ is harmonic. Using the continuous representatives in the pointwise hypothesis makes this conclusion pointwise, rather than only almost everywhere. This is compact convergence of locally bounded harmonic functions.