1549901369-Elements_of_Real_Analysis__Denlinger_

B.3 Algebra of Real-Valued Functions 631 Definition B.3.8 Let A denote an arbitrary set. The identity function on A is the funct ...

632 Appendix B • Sets and Functions Next, the ~ direction. Suppose f : A----; B is 1-1 and onto B. Let b E B. Since f is onto B, ...

B.3 Algebra of Real-Valued Functions 633 10. Prove Theorem B.3.3, (11). Prove Theorem B.3.3, (14). 12. Prove Theorem B.3.3, (1 ...

...

Appendix C Answers & Hints for Selected Exercises Caveat: The answers and hints provided here are intentionally brief. Compl ...

636 Appendix C 11 Answers & Hints for Selected Exercises Using (h) and (M2), (-x)y = [(-l)x]y = (-l)(xy) = -(xy). Similarly ...

Answers & Hints for Selected Exer cises 637 Case 2: a = b-/2 => a = b = 0, since otherwise % = -/2, which would tell us t ...

638 Appendix C 11 Answers & Hints for Selected Exercises k(3k 2 -1) + 3k + 1 -- 3k^2 +5k+2 2 - - (k+l)(3k+2) 2 - - (k+1)( ...

Answers & Hints for Selected Exercises 639 by Thm. 1.5.3. Suppose x < 0. Then -x > 0 so by Thm. 1.5.3, :Jm E N 3 m - 1 ...

640 Appendix C • Answers & Hints for Selected Exercises (b) If v is any lower bound for A, then by Exercise 1.6-A. 12 , - v ...

Answers & Hints for Selected Exercises 641 (1) Using algebra and Thm. 2.2.13, 1 . 4n^2 - 5n 4 - .2. im = lim n n->= 8n^2 ...

642 Appendix C 11 Answers & Hints for Selected Exercises Xn = ~ -; 0. (Apply Exercise 2.2.16.) When lei < 1, nP I Xn+l \ ...

Answers & Hints for Selected Exercises 643 n! 1 n 100 I Yn+l I (n + 1) 100 (c) Let Xn = n! (^100) n and Yn = -Xn = -n. 1 -. ...

644 Appendix C • Answers & Hints for Selected Exercises Modify the proof of (a), replacing "increasing" by "decreasing," "s ...

Answers & Hints for Selected Exercises 645 Use induction to show that the sequence is monotone increasing and bounded above ...

646 Appendix C • Answers & Hints for Selected Exercises EXERCISE SET 2. 7 (^1) • (a) m n ' > l g =? 1 i m - .l n I < m ...

Answers & Hints for Selected Exercises 64 7 EXERCISE SET 2.8 Let A = { ai, a2, · · · , an · · · } and B = {bi, b2, · · · , ...

648 Appendix C • Answers & Hints for Selected Exercises (a) 0, -1 (f) 0, 0 (b) 1, -1 (g) 0, 1, +oo EXERCISE SET 2.9 (c) +o ...

Answers & Hints for Selected Exercises 649 (==>) Suppose A is open. Let x EA. Then :le> 0 3 Nc;(x) ~A, so Nc;(x) cont ...

650 Appendix C • Answers & Hints for Selected Exercises (a) Let x E R Since A has only finitely many elements, x cannot sati ...