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
Chapter2: The Integers
Section2.1: Postulates For The Integers (optional)
Problem 29E
Related questions
Question
please show work
![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.]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F912a24df-6035-40a5-a35d-878c2e7f673f%2F07a42799-71e4-418f-aaf5-f788592372d4%2Fmqnf7ho_processed.png&w=3840&q=75)
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.]
Expert Solution

This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 5 steps with 5 images

Recommended textbooks for you

Elements Of Modern Algebra
Algebra
ISBN:
9781285463230
Author:
Gilbert, Linda, Jimmie
Publisher:
Cengage Learning,
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage

Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning

Elements Of Modern Algebra
Algebra
ISBN:
9781285463230
Author:
Gilbert, Linda, Jimmie
Publisher:
Cengage Learning,
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage

Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning

College Algebra (MindTap Course List)
Algebra
ISBN:
9781305652231
Author:
R. David Gustafson, Jeff Hughes
Publisher:
Cengage Learning

Elementary Linear Algebra (MindTap Course List)
Algebra
ISBN:
9781305658004
Author:
Ron Larson
Publisher:
Cengage Learning