Let $n\ge 2$ and suppose that $z_1, z_2, \ldots, z_n$ are distinct points in the interior of some disk $D$ in the plane. Why is it true that there exists a smaller disk $D'\subseteq D$ such that $D'$ contains exactly $n-1$ of these points above?
In other words, we want $D'$ to contain $n-1$ of the points $z_1, z_2, \ldots, z_n$, and miss one of these points. This fact above was used in the proof of Cauchy's Residue Theorem in my complex analysis class. However, the proof wasn't provided, and the justification for it was kind of handwaving. I would love to see a rigorous proof for this interesting fact.
Any help is much appreciated!
P.S. What area of mathematics would this problem fall? (Topology? Geometry?)