Let $z_{i}$ be a complex number. Prove that a finite set of points $z_{1}$…… $z_{n}$ cannot have any accumulation points.
What i tried.
Proving by contradiction Let $S$ be the set in the complex plane. I suppose that there is at least one accumulation point $z_{o}$. Then by the definition of accumulation point, we know that the neighborhood of $z_{o}$ must have at least a point that lies in $S$ and those points are $z_{1}$…… $z_{n}$ as given in the question. To arrive at a contradiction, I must show that there is indeed an infinite number of such points and thus infinite number of distances between these points and the accumulation point. I tried using the Eplison delta proof to show this but im stuck at the Eplison delta portion. Would anyone be able to explain the eplison delta proof? Thanks
Best Answer
For $z_0$ to be an accumulation point, we must have that every neighborhood of $z_0$ has to contain a point from your set different from $z_0$. Using the fact that you have a finite set, try to come up with a neighborhood around $z_0$ that will not contain any of the $z_1,z_2,\dots,z_n$. That is, try to come up with an appropriate $\varepsilon$ radius for your neighborhood around $z_0$.