*7. Let S be a nonempty bounded subset of R and let ke R. Define kS = {ks : se S}. Prove the following: (a) Ifk20, then sup (kS) = k · sup S and inf (kS) = k - inf S. (b) If k<0, then sup (kS) = k - inf S and inf (kS) = k - sup S.
*7. Let S be a nonempty bounded subset of R and let ke R. Define kS = {ks : se S}. Prove the following: (a) Ifk20, then sup (kS) = k · sup S and inf (kS) = k - inf S. (b) If k<0, then sup (kS) = k - inf S and inf (kS) = k - sup S.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![The Real Numbers
*7. Let S be a nonempty bounded subset of R and let ke R. Define kS =
{ks :se S}. Prove the following:
(a) If k20, then sup (kS) = k · sup S and inf (kS) =k · inf S.
(b) Ifk< 0, then sup (kS) = k · inf S and inf (kS) =k· sup S.
8. Let S and T be nonempty bounded subsets of R with S T. Prove that
inf T< inf S < sup S < sup T.
9. (a) Prove: If y > 0, then there exists ne N such that n-1< y < n. ☆
(b) Prove that the n in part (a) is unique.
10. (a) Prove: If x and y are real numbers with x < y, then there are infinitely
many rational numbers in the interval [x, y].
(b) Repeat part (a) for irrational numbers.
11. Let y be a positive real number. Prove that for every n e N there exists a
I unique positive real number x such that x" = y. ☆
*12. Let D be a nonempty set and suppose that f: D → R and g: D → R. Define
the function f +g:D→R by (f+g)(x)= f(x) +g(x).
(a) If f(D) and g(D) are bounded above, then prove that (f + g)(D) is
bounded above and sup[(ƒ +g)(D)] < sup f(D) + sup g(D).
(b) Find an example to show that a strict inequality in part (a) may occur.
(c) State and prove the analog of part (a) for infima.
13. Let x e R. Prove that x = sup {q e Q:q< x}. ☆
14. Let a/b be a fraction in lowest terms with 0< a/b<1.
(a) Prove that there exists ne N such that
1
a
<
1
n+1
b
(b) If n is chosen as in part (a), prove that a/b– 1/(n+1) is a fraction that in
lowest terms has a numerator less than a.
(c) Use part (b) and the principle of strong induction (Exercise 1.27) to
prove that a/b can be written as a finite sum of distinct unit fractions:
1
1
a
%3D
b
where n, ..., nɛ e N. (As a point of historical interest, we note that in
the ancient Egyptian system of arithmetic all fractions were expressed as
sums of unit fractions and then manipulated using tables.)
15. Prove Euclid's division algorithm: If a and b are natural numbers, then there
exist unique numbers q and r, each of which is either 0 or a natural number,
such that r<a and b = qa + r. ☆](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3bf7da54-5138-4b57-a0f9-aec30fa8e8cf%2Fdd585b7b-4cac-408b-b5e9-e4d06133b0bc%2Fspslqro_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The Real Numbers
*7. Let S be a nonempty bounded subset of R and let ke R. Define kS =
{ks :se S}. Prove the following:
(a) If k20, then sup (kS) = k · sup S and inf (kS) =k · inf S.
(b) Ifk< 0, then sup (kS) = k · inf S and inf (kS) =k· sup S.
8. Let S and T be nonempty bounded subsets of R with S T. Prove that
inf T< inf S < sup S < sup T.
9. (a) Prove: If y > 0, then there exists ne N such that n-1< y < n. ☆
(b) Prove that the n in part (a) is unique.
10. (a) Prove: If x and y are real numbers with x < y, then there are infinitely
many rational numbers in the interval [x, y].
(b) Repeat part (a) for irrational numbers.
11. Let y be a positive real number. Prove that for every n e N there exists a
I unique positive real number x such that x" = y. ☆
*12. Let D be a nonempty set and suppose that f: D → R and g: D → R. Define
the function f +g:D→R by (f+g)(x)= f(x) +g(x).
(a) If f(D) and g(D) are bounded above, then prove that (f + g)(D) is
bounded above and sup[(ƒ +g)(D)] < sup f(D) + sup g(D).
(b) Find an example to show that a strict inequality in part (a) may occur.
(c) State and prove the analog of part (a) for infima.
13. Let x e R. Prove that x = sup {q e Q:q< x}. ☆
14. Let a/b be a fraction in lowest terms with 0< a/b<1.
(a) Prove that there exists ne N such that
1
a
<
1
n+1
b
(b) If n is chosen as in part (a), prove that a/b– 1/(n+1) is a fraction that in
lowest terms has a numerator less than a.
(c) Use part (b) and the principle of strong induction (Exercise 1.27) to
prove that a/b can be written as a finite sum of distinct unit fractions:
1
1
a
%3D
b
where n, ..., nɛ e N. (As a point of historical interest, we note that in
the ancient Egyptian system of arithmetic all fractions were expressed as
sums of unit fractions and then manipulated using tables.)
15. Prove Euclid's division algorithm: If a and b are natural numbers, then there
exist unique numbers q and r, each of which is either 0 or a natural number,
such that r<a and b = qa + r. ☆
Expert Solution
![](/static/compass_v2/shared-icons/check-mark.png)
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 2 steps
![Blurred answer](/static/compass_v2/solution-images/blurred-answer.jpg)
Recommended textbooks for you
![Advanced Engineering Mathematics](https://www.bartleby.com/isbn_cover_images/9780470458365/9780470458365_smallCoverImage.gif)
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
![Numerical Methods for Engineers](https://www.bartleby.com/isbn_cover_images/9780073397924/9780073397924_smallCoverImage.gif)
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
![Introductory Mathematics for Engineering Applicat…](https://www.bartleby.com/isbn_cover_images/9781118141809/9781118141809_smallCoverImage.gif)
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
![Advanced Engineering Mathematics](https://www.bartleby.com/isbn_cover_images/9780470458365/9780470458365_smallCoverImage.gif)
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
![Numerical Methods for Engineers](https://www.bartleby.com/isbn_cover_images/9780073397924/9780073397924_smallCoverImage.gif)
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
![Introductory Mathematics for Engineering Applicat…](https://www.bartleby.com/isbn_cover_images/9781118141809/9781118141809_smallCoverImage.gif)
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
![Mathematics For Machine Technology](https://www.bartleby.com/isbn_cover_images/9781337798310/9781337798310_smallCoverImage.jpg)
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
![Basic Technical Mathematics](https://www.bartleby.com/isbn_cover_images/9780134437705/9780134437705_smallCoverImage.gif)
![Topology](https://www.bartleby.com/isbn_cover_images/9780134689517/9780134689517_smallCoverImage.gif)