1549901369-Elements_of_Real_Analysis__Denlinger_

To the Student xxi take a critical, even skeptical approach. Mere passive acceptance will not do! In fact, we will be so critica ...

xxii To the Student You will recall that calculus is built upon the concept of limit. Thus, Chap- ter 2 takes up that idea. Alon ...

To the Student xx iii statements and proofs by calling the reader's attention to the presence of for- mal logical patterns. In p ...

xxiv To the Student this book as you would read a newspaper or a novel. You must "read" this book with pencil and paper. Write o ...

To the Instructor When Newton and Leibnii3 invented the calculus in the late sev- enteenth century, they did not use delta-epsil ...

xxvi To the Instructor In her essay [56] quoted at the beginning of this section, Grabiner points to another factor contributing ...

To the Instructor xx vii But it causes the student to recognize the importance of the hypotheses, and to rethink what they actua ...

xx viii To the Instructor definition. Its presence is crucial, for example, in the proof of Theorem 4.1.9, the sequential criter ...

To the Instructor xx ix Section(s) Description Number of Days (Core) (Optional) 1.1-1.4 Fields; ordered fields; natural and rati ...

xxx To the Instructor 7.1 Refresher on suprema & infima. 0 .5 7.2 R iemann integral defined via Darboux sums. 2 7.3 The inte ...

Chapter 1 The Real Number System Sections 1.1-1.4 are optional, containing background on or- dered fields and the rational numbe ...

2 Chapter 1 • The Real Number System efficient and honest. The constructive approach would detain us too long , and the pragmati ...

1.1 The Field Properties 3 A more complete discussion of these logical symbols and others, along with important rules for their ...

4 Chapter 1 • The Real Number System Some number systems are fields, but many are not. The following exercises will help you see ...

1.1 The Field Properties 5 The set {O, 1, 2, 3} with the operations + and · defined by the tables: +^0 1 2 3 0 1 2 3 0 0 1 2 3 ...

6 Chapter 1 11 The Real Number System Proof. (a) Suppose y + x = z + x. Then, using (A3) and (A4), :Ju E F 3 x + u = 0, and (giv ...

(a) -0 = O; (b) 't:/x E F, -(-x) = x; (c) 1-^1 = 1 and (-1)-^1 = -1; ( d) 't:/x E F, x · 0 = O; (e) xy=O<=:? eitherx=O ory=O; ...

8 Chapter 1 • The Real Number System Thus, x -j. 0 ::::} y = O; that is, either x = 0 or y = 0. Part 2: For the "~" part, observ ...

1.1 The Field Properties 9 (d) Exercise 12. • Theorem 1.1. 8 (Properties of Division and Fractions) In any fi eld F , (a) Vx E F ...

10 Chapter 1 • The Real Number System (h) Exercise 18. (i) Exercise 1 9. (j) Exercise 20. • EXERCISE SET 1.1-B Prove Theorem 1. ...