I have to prove that if $\sum\limits_{n=1}^\infty \frac{1}{|a_n|}$ converges, then $\sum\limits_{n=1}^\infty \frac{1}{x-a_n}$ converges absolutely and uniformly in any compact space that $\forall n \in \mathbb N$ doesn't contain point $x=a_n$.
Since there is a compact space and all functions $\frac{1}{x-a_n}$ are continuous on it, maybe it's worth to use Dini's theorem, but I can't figure out how to apply it, because I need to show that this series converges pointwise in this compact space.