1549901369-Elements_of_Real_Analysis__Denlinger_

l. 2 The Order Properties 11 1.2 The Order Properties In working with real numbers we make frequent use of the concepts of less ...

12 Chapter 1 • The Real Number System ( c) x :::; y iff x 1 y; x ::::: y iff x </. y. (d) Ifx:s;yandy:s;x,thenx=y. Proof. (a) ...

1.2 The Order Properties 13 Theorem 1.2.8 (Algebraic Properties of Inequalities) For any ordered field F, the following properti ...

14 Chapter 1 • The Real Number System Theorem 1.2.10 (Further Algebraic Properties of Inequalities) In any ordered field F , the ...

1.2 The Order Properties 15 ( c) Suppose F is an ordered field. Then 1 E F and, since F is closed under addition, F must contain ...

16 Chapter 1 • The Real Number System Prove the second claim of Theorem 1.2.10 (c). Prove Theorem 1.2.10 (d). Prove Theorem 1.2 ...

Proof. (a) Exercise l. (b) By Definition 1.2.12, I -x f = { -x if -(-x) if 1.2 The Order Properties 17 x20 x<O { x if - ( ...

18 Chapter 1 • The Real Number System Theorem 1.2.15 (Triangle Inequalities) For all x, y in an ordered field F, (a) Ix+ YI :::: ...

1.2 The Order Properties 19 (c) (a, b] = {x E F: a< x:::; b}; (d) [a, b) = {x E F: a:::; x < b}; (e) (- oo,b) = {x E F: x ...

20 Chapter 1 11 The Real Number System (b) Three positive elements a, b, c of an ordered field form a geometric b c progression ...

1.3 Natural Numbers 21 Theorem 1.3.2 The intersection^5 of any collection of inductive subsets of F is inductive. Proof. Let C d ...

22 Chapter 1 • The Real Number System Therefore, A is an inductive set. By Theorem 1.3.4, Np ~ A. That is, all natural numbers i ...

1.3 Natural Numbers 23 Then: (1) p(l) is true, since ~m E NF 3 m < l. (2) Suppose p(k) is true. That is, m < k::::} k - m ...

24 Chapter 1 • The Real Number System (b) Let n E NF be fixed. Vm E NF we let p(m) denote the proposition p(m): nm E NF. Then: ( ...

1.3 Natural Numbers 25 *Proof. Suppose p(n) satisfies c~nditions (1) and (2) above. Let q(l) de- note p(l), and for k ~ 2 let q( ...

26 Chapter 1 • The Real Number System *Proof. Exercise 18. • Similarly, the Alternate Principle of Mathematical Induction can be ...

1.3 Natural Numbers 27 12. Finite Geometric Sums: If r-:/:-1, a+ ar + ar^2 + ar^3 + · · · + arn = a - arn+l 1-r 13. 2n :S (n + 1 ...

28 Chapter 1 • The Real Number System 1.4 Rational Numbers We have seen in Section 1.3 that every ordered field contains the nat ...

1.4 Rational Numbers 29 (MO) Let x,y E Qp. Then 3m,n,m',n' E Zp 3 n , n' I= 0, x =~ and y -- m' ti!• Th us, mm' mm' x · y = - · ...

30 Chapter 1 • The Real Number System In this sense, every ordered field contains the familiar natural numbers, integers, and ra ...