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Exercise 6.41. Let E be a nonempty set in Rn. Then the distance of a point x € R¹ to E is defined by dist (x, E) = inf {||x - y|| : y € E}. Prove the following: (i) x E E if and only if dist (x, E) = 0. (ii) E is closed if and only if dist (x, E) > 0 for all x € E. (iii) For each x ER", there exists a point y in E such that dist (x, E) = ||x - y||.

Exercise 6.41. Let E be a nonempty set in Rn. Then the distance of a point x € R¹ to E is defined by dist (x, E) = inf {||x - y|| : y € E}. Prove the following: (i) x E E if and only if dist (x, E) = 0. (ii) E is closed if and only if dist (x, E) > 0 for all x € E. (iii) For each x ER", there exists a point y in E such that dist (x, E) = ||x - y||.

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter5: Rings, Integral Domains, And Fields
Section5.4: Ordered Integral Domains
Problem 8E: If x and y are elements of an ordered integral domain D, prove the following inequalities. a....
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Exercise 6.41. Let E be a nonempty set in Rn. Then the distance of a point x E R to E is defined by dist (x, E) = inf {||x - y|| : y = E}. Prove the following: (i) x E E if and only if dist (x, E) = 0. (ii) E is closed if and only if dist (x, E) > 0 for all x € EC. (iii) For each x ER", there exists a point y in E such that dist (x, E) = |x-y||.
Transcribed Image Text:Exercise 6.41. Let E be a nonempty set in Rn. Then the distance of a point x E R to E is defined by dist (x, E) = inf {||x - y|| : y = E}. Prove the following: (i) x E E if and only if dist (x, E) = 0. (ii) E is closed if and only if dist (x, E) > 0 for all x € EC. (iii) For each x ER", there exists a point y in E such that dist (x, E) = |x-y||.
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