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
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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.]](/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.]
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