This could be something which is already somewhere in the website, but I am unable to locate any.
Prove $\prod_{n=1}^{\infty} (1-z^n)$ converges absolutely and uniformly on each compact subset of $\biggr[{|z|<1}\biggr]$.
What about $\biggr[{|z|>1}\biggr]$?
I have a feeling this need to be done with using some logarithm expansion and using Weierstrass theorem. But I do like to see more ideas of proof. I am not sure on using those ideas either.