Here's the exercise:
Let $\delta\in(0,1)$ and let $(a_n)_{n\in\mathbb{N}}$ be a real, monotonic decreasing sequence that converges to $0$. Show that $\sum a_nz^n$ converges uniformly on $\{|z|\leq1\}\cap\{|z-1|>\delta\}$
I quite frankly don't have any idea of how to approach this. Any hints and nudges in the correct direction are greatly appreciated.