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I have no idea where to start with this one, as the possibilities of where to start seem infinite. I'm not sure if elementary complex analysis will suffice, or more advanced techniques are needed. To put it into context, this is exercise 23.2 of Otto Forster's Letures on Riemann Surfaces (the exercise appears after he talks about Dirichlet boundary value problem, subharmonic/harmonic functions and introduced the notion of a "Runge subset" of a Riemann surface. However Runge's theorem certainly hasn't been introduced yet).

$Y \subset \mathbb{C}$ is open and $K$ is a compact connected component of $\mathbb{C} \setminus Y$. $(f_n)$ is a sequence of polynomials which converges uniformly on every compact subset of $Y$. I want to show that $(f_n)$ converges uniformly on $K$.

Could I please get some hints/ guidance?

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Maximum principle should do it. Think of a contour $\gamma$ within $Y$ that surrounds $K$. Uniform convergence on $\gamma$ implies uniform convergence on the domain bounded by $\gamma$, in particular on $K$.

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  • $\begingroup$ Thank you for your help! It's clear now, but if I wanted to make everything precise, I'd need to show (properly!) that K can be surrounded by that contour $\gamma$ within $Y$. That seems like a topological argument - is it complicated? Just having a bit of a brain-dead moment... $\endgroup$ – Acton Mar 13 '18 at 7:23

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