5. (P19, Page 12) Define S = {r € R | ² < x}. Do the following. (a) Use properties of real numbers to prove that 1 is an upper bound for S. [Properties needed: (i) If a ± 0, a² > 0. (ii) If a > 0, then a¯1> 0. (iii) If a < b and 0 < c, then ac < bc.]
5. (P19, Page 12) Define S = {r € R | ² < x}. Do the following. (a) Use properties of real numbers to prove that 1 is an upper bound for S. [Properties needed: (i) If a ± 0, a² > 0. (ii) If a > 0, then a¯1> 0. (iii) If a < b and 0 < c, then ac < bc.]
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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![5. (P19, Page 12) Define S = {x € R | x² < x}. Do the following.
Use properties of real numbers to prove that 1 is an upper bound for S.
(a)
[Properties needed: (i) If a ± 0, a² > 0. (ii) If a > 0, then a- > 0. (iii) If a < b and 0 < c, then ac < bc.]
(b)
Use the Method of Proof by Contradication to prove that sup S = 1.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F072e444d-1bde-4899-b3c3-9f07885f3d58%2Fccd91a3e-399f-4081-bffa-7df2da183f76%2Fqejkbyol_processed.png&w=3840&q=75)
Transcribed Image Text:5. (P19, Page 12) Define S = {x € R | x² < x}. Do the following.
Use properties of real numbers to prove that 1 is an upper bound for S.
(a)
[Properties needed: (i) If a ± 0, a² > 0. (ii) If a > 0, then a- > 0. (iii) If a < b and 0 < c, then ac < bc.]
(b)
Use the Method of Proof by Contradication to prove that sup S = 1.
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