TITLE: a holomorphic function is uniform limit of polynomials
QUESTION [5 upvotes]: I was reviewing my complex analysis, and found this problem in a problem set. It says "prove that every holomorphic function on the disc $D=\{|z|<1\}$ is a uniform limit of polynomials". 
I'm confused about it, it seems to me that the statement as it is is not true. It should be true that the convergence is uniform in every compact set contained in the disc though. However, for instance, I think it is not possible to approximate the function $\frac{1}{z-1}$ with polynomials that converge uniformly. Am I right?

REPLY [10 votes]: Of course you are right: it is impossible to approximate an unbounded function uniformly with bounded functions.