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Let $(f_n)_{n\in\mathbb{N}}$ be a sequence of entire functions and let $f$ be an entire function such that, for every compact subset $K$ of the unit disk, it holds that: $$\|f_n-f\|_{L^\infty(K)}\rightarrow0, n\rightarrow\infty.$$ Is it true that if $H$ is a compact subset of $\mathbb{C}$ then $$\|f_n-f\|_{L^\infty(H)}\rightarrow0, n\rightarrow\infty?$$ My guess is that the answer is no, and probably the counterexample is obvious, but I'm having hard time finding it.

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$f_n(z)=(\frac z 2)^{n}$ is a counter-example. Here $f=0$ and you don't have convegrence when $H=\{3\}$.

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  • $\begingroup$ Why did you divide by $2$? $\endgroup$ – zhw. Jul 19 '18 at 14:53
  • $\begingroup$ Silly of me! I was thinking of convergence on the closed disk $\endgroup$ – Kavi Rama Murthy Jul 19 '18 at 23:34

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