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IMC 2003 Problem 9

Source: IMC 2003 Day 2 Problem 3

November 2, 2020
densetopologyreal analysiscollege contestsIMC

Problem Statement

Let AA be a closed subset of Rn\mathbb{R}^{n} and let BB be the set of all those points b∈Rnb \in \mathbb{R}^{n} for which there exists exactly one point a0∈Aa_{0}\in A such that ∣a0−b∣=inf⁡a∈A∣a−b∣|a_{0}-b|= \inf_{a\in A}|a-b|. Prove that BB is dense in Rn\mathbb{R}^{n}; that is, the closure of BB is Rn\mathbb{R}^{n}
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