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analysis

Source: miklos schweitzer 1995 q10

October 2, 2021
geometryreal analysis

Problem Statement

Let X={X1,X2,...}X =\{ X_1 , X_2 , ...\} be a countable set of points in space. Show that there is a positive sequence {ak}\{a_k\} such that for any point Z∉XZ\not\in X the distance between the point Z and the set {X1,X2,...,Xk}\{X_1,X_2 , ...,X_k\} is at least aka_k for infinitely many k.
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