Exercise 6.53. Let K be a closed and bounded subset of an open set U in R". Provethat there exists an r> 0 such that B, (a) C U for all a E K, by applying each of thefollowing compactness results:(i) the Bolzano-Weierstrass theorem.(ii) the Heine-Borel theorem.

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Answered: Exercise 6.53. Let K be a closed and…

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter7: Distance And Approximation

Section7.1: Inner Product Spaces

Problem 16EQ

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Exercise 6.53. Let K be a closed and bounded subset of an open set U in R". Prove that there exists an r> 0 such that B, (a) C U for all a € K, by applying each of the following compactness results: (i) the Bolzano-Weierstrass theorem. (ii) the Heine-Borel theorem.