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I'm studying for quals, and one question is:

Let $u_n, n=1,2,\dots$ be a sequence of harmonic functions in a complex domain $\Omega$ such that $|u_n(z)| < C$ in $\Omega$ for some $C>0$ and all $n$. Suppose that there exists a subdomain $\Gamma \subset \Omega$ such that the sequence $u_n(z)$ converges for any $z\in\Gamma$. Prove that then $u_n(z)$ converges for any $z\in\Omega$ and the limit function is harmonic in $\Omega$.

It looks like normal convergence thing. My idea is that put them in analytic function and use Montel's Theorem. But the approach seems weird to me.

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