Linear Algebra: A Modern Introduction
4th Edition
ISBN: 9781285463247
Author: David Poole
Publisher: Cengage Learning
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Question
11
![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. ☆](https://content.bartleby.com/qna-images/question/3bf7da54-5138-4b57-a0f9-aec30fa8e8cf/f2fb7476-66db-4124-be44-792447b5f149/p2ne6gp_processed.jpeg)
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. ☆
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