6. The Archimedean Property of the real numbers is the following statement: For every x, y ER such that x > 0 and y > 0, there exists n EN such that nx > y. (a) Prove that for every a ER, there exists ne N such that n > a. That is, N is not bounded above. [HINT: Proceed by contradiction, and use the Least Upper Bound Property of R.] (b) Use part (a) to prove that the Archimedean Property is true. (c) Use the Archimedean Property to prove that for every x € R such that x > 0, there exists n E N such that ⁄ < x. 1 n
6. The Archimedean Property of the real numbers is the following statement: For every x, y ER such that x > 0 and y > 0, there exists n EN such that nx > y. (a) Prove that for every a ER, there exists ne N such that n > a. That is, N is not bounded above. [HINT: Proceed by contradiction, and use the Least Upper Bound Property of R.] (b) Use part (a) to prove that the Archimedean Property is true. (c) Use the Archimedean Property to prove that for every x € R such that x > 0, there exists n E N such that ⁄ < x. 1 n
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter1: Fundamentals
Section1.7: Relations
Problem 21E: 21. A relation on a nonempty set is called irreflexive if for all. Which of the relations in...
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