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Let $\Omega$ be a bounded connected open subset of $\mathbb C$. Let $\{f_n\}$ be sequence of functions which are continuous on $\bar{\Omega}$ and holomorphic on $\Omega$. Assume that the sequence converges uniformly on the boundary of $\Omega$ .Show that $\{f_n\}$ converges uniformly on $\bar \Omega$ to a function which is continuous on $\bar \Omega$ and holomorphic on $\Omega$.

I was using Weierstrass theorem(complex analysis) in which the sequence should have to be convergent uniformly on every compact subset of $\Omega.$ But here I can not apply it.

Please someone give some hints..

Thank you..

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By the maximum modulus principle, we have

$$ \sup_{z \in \overline{\Omega}} |f_n(z) - f_m(z)| = \sup_{z \in \partial \Omega} |f_n(z) - f_m(z)|. $$

Since $f_n$ converges uniformly on $\partial \Omega$ to some function, the sequence $(f_n)$ is uniformly Cauchy on $\partial \Omega$ so the right hand side tends to zero as $m,n \to \infty$. This implies that $(f_n)$ is uniformly Cauchy on $\overline{\Omega}$ so $f_n$ converge uniformly to some continuous function $f$ on $\overline{\Omega}$. In particular, $f_n$ also converges uniformly to $f$ on every compact subset of $\Omega$ so $f$ is holomorphic on $\Omega$.

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