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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
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter2: The Integers
Section2.1: Postulates For The Integers (optional)
Problem 29E
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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 € R, there exists n ≤ 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 EN such that < x. n (d) Prove that there is a rational number between any two real numbers. That is, for every a, b R with a ≤ b, there exists q E Q such that a <q < b. [HINT: Start by using part (c) to find a denominator for q. Then, use the Well-Ordering Axiom to choose a numerator for q.]
Transcribed Image Text: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 € R, there exists n ≤ 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 EN such that < x. n (d) Prove that there is a rational number between any two real numbers. That is, for every a, b R with a ≤ b, there exists q E Q such that a <q < b. [HINT: Start by using part (c) to find a denominator for q. Then, use the Well-Ordering Axiom to choose a numerator for q.]
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