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This is an exercise from John Conway's book on complex analysis:

Investigate if there exists a sequence of polynomials $(P_n)$ that fulfills the conditions $P_{n}(0)=1$ for all natural numbers $n$ and $\lim_{n\rightarrow\infty}P_n(z)=0$ for all $z\neq0$

Polynomials obey the maximum principle, but I don't see how to apply it if all we know is point-wise convergence. (Uniform convergence would imply $|P_n|<\epsilon$ on the unit circle for large $n$, contradicting $P_n(0)=1$.)

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    It was a bad idea...2012-07-02

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I think this can be done using Mergelyan's theorem: Let $K_n$ be a compact set of the form $\{0\}\cup \{z: 1/n\le |z|\le n, |\arg z-\theta_n|\ge \epsilon_n \}$. On this set you can approximate the function $\chi_{\{0\}}$ by a polynomial $P_n$ within $1/n$. Make sure to choose $\theta_n$ and $\epsilon_n$ so that no point of the plane belongs to $K_n^c$ for infinitely many $n$.

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    @ec52 Divide by $p_n(0)$.2012-07-07