Let $\{p_n\}$ be a set of (complex) polynomials such that $$ \displaystyle \lim_{n\to \infty} p_n(z) =1$$, for all $z$ on the unit circle $C(0,1)$ (uniformly). How to show that this also holds (using the maximum modulus principle) that on the unit disc $D(0,1)$?
Approach: I've no idea how to start. I don't see in particular how I can use the max mod principle in this problem, so any hints/solutions are welcome. Thanks in advance!
EDIT: the maximum modulus principle says that a non constant holomorphic function (in a region) cannot attain it's maximum in that region.