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- 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 [(f +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.

- 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 [(f +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.

Advanced Engineering Mathematics
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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12b,c
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. ☆
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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