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Since D' is bounded and closed, by the Heine-Borel Theorem, D' is compact. In D', $A_k$ would have an accumulation point. How can I extended that into D? How does that help with showing $f$ is a polynomial?Take \(D'\subset D\) any closed subdisk, and consider the sets \(A_n=\{ x\in D' : f^{(n)}(x)=0 \}\). Prove that one of these, say \(A_k\), has an accumulation point in \(D\), what can you say about \(f^{(k)}\)?

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We have a sequence $a_m\in A_m$ in $D'$. By the Heine-Borel Theorem, $D'$ is compact and has a convergent subsequence of $a_m$. Therefore, $f^{(m)} = 0$ and $f$ is a polynomial.

Is this good?

As long as you know why each of your claims is valid then yes, everything's fine.

We have a sequence $a_m\in A_m$ in $D'$. By the Heine-Borel Theorem, $D'$ is compact and has a convergent subsequence of $a_m$. Therefore, $f^{(m)} = 0$ and $f$ is a polynomial.

Is this good?

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I think I am unsure of is $f^{m} = 0$ and $f$ is a polynomial. Can you explain why that is the case?As long as you know why each of your claims is valid then yes, everything's fine.