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 $P_n(0)=1$.)