1549901369-Elements_of_Real_Analysis__Denlinger_

1.5 The Archimedean Property 31 1.5 The Archimedean Property In this section, we discuss a property so natural that it may seem ...

32 Chapter 1 • The Real Number System Theorem 1.5.3 Every positive element of an Archimedean ordered field can be located betwee ...

1.5 The Archimedean Property 33 DENSE SETS IN ORDERED FIELDS Definition 1.5.6 A set S is dense in an ordered field F if Va< b ...

34 Chapter 1 l'!I The Real Number System Proof. (a) Suppose that Ve: > 0, x:::::; c:. For contradiction, suppose x 1:. 0. The ...

1.6 The Completeness Property 35 where ao, ai,. · ·an are (constant) real numbers. If an -/=-0, then n is called the "degree" of ...

36 Chapter 1 • The Real Number System If A has an upper bound we say that A is bounded above; if A has a lower bound we say that ...

1.6 The Completeness Property 37 are not minimum or maximum elements of (a, b), there is something special about a and b. In tec ...

38 Chapter 1 • The Real Number System be members of A, inf A and sup A do not necessarily pelongj;o A. In fact, we must be very ...

1.6 The Completeness Property 39 (i) Let x E A. By (a), Ve > 0, x < u + e. Thus, by the "forcing principle" [Theorem 1.5.9 ...

40 Chapter 1 • The Real Number System For each of the sets listed in Exercise 1, tell whether or not the given set is bounded b ...

1.6 The Completeness Property 41 Theorem 1.6.9 Any complete ordered field is Archimedean. Proof. Suppose F is a complete ordered ...

42 Chapter 1 • The Real Number System But we also know from Theorem 1.2.8 (e) that (u-8)^2 <u^2 < (u+o)^2. Adding these tw ...

1.6 The Completeness Property 43 Definition 1.6.14 (-oo and +oo as infimum and supremum): Let JR denote the complete ordered fie ...

44 Chapter 1 • The Real Number System Let X be a nonempty set and suppose f, g : X ---+ IR are functions whose ranges are bound ...

7 *"The" Complete Ordered Field 45 classic reference^10 is [82]; other constructions may be found in Chapter 1 of Part B of [1 ...

46 Chapter 1 • The Real Number System (4) f(lp) = lpt; (5) Vx E F, f(-x) = -f(x); (6) Vx E F, f(x-^1 ) = f(x)-^1 ; (7) For each ...

7 * "The" Complete Ordered Field 4 7 Lemma 1.7.2 If Fis a complete ordered field, then Vx E F, x = sup{ r E Qp : r < x}. Th ...

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Chapter 2 Sequences Sections 2.1- 2.7, through Theorem 2.7.4, contain essential core material. Indeed, the concepts and methodol ...

50 Chapter 2 • Sequences 2.1 Basic Concepts: Convergence and Limits A commonly-held, intuitive understanding of an (infinite) se ...