Sequence of polynomials converging to zero, but not uniformly on unit disc

Question

I have been trying to solve the following without success so far:

Show that there exists a sequence of polynomials satisfying $P_n(z)\rightarrow 0$ for every $z\in \mathbb{C}$, but the convergence is not uniform on the unit disc $D$.

I know i am supposed to use Runge's theorem, i have seen similar exercises in which the plane is exhausted using compact sets with connected complement, while approximating the desired function step by step, but the uniform convergence bit in this one is where i got stuck. Could it be something like the argument principle? Since we don't know how many roots those polynomials have in the disc, maybe a function with infinite zeroes does the trick, the zero function for example?

I will appreciate all help.

Answer 1

Hint: As long as there are $z_n \in D$ with $|f_n(z_n)| \ge 1$, say, the convergence can't be uniform in $D$.