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The Real Numbers: Sets, Sequences, and Functions

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The Real Numbers: Sets, Sequences, and Functions CHAPTER 1 The Real Numbers: Sets, Sequences, and Functions Contents 1.1 The Field, Positivity, and Completeness Axioms . 7 1.2 The Natural and Rational Numbers . 11 1.3 Countable and Uncountable Sets . 13 1.4 Open Sets, Closed Sets, and Borel Sets of Real Numbers . 16 1.5 Sequences of Real Numbers . 20 1.6 Continuous Real-Valued Functions of a Real Variable . 25 We assume the reader has a familiarity with the properties of real numbers, sets of real numbers, sequences of real numbers, and real-valued functions of a real variable, which are usually treated in an undergraduate course in analysis. This familiarity will enable the reader to assimilate the present chapter, which is devoted to rapidly but thoroughly establishing those results which will be needed and referred to later. We assume that the set of real numbers, which is denoted by R, satisfies three types of axioms. We state these axioms and derive from them properties on the natural numbers, rational numbers, and countable sets. With this as background, we establish properties of open and closed sets of real numbers; convergent, monotone, and Cauchy sequences of real numbers; and continuous real-valued functions of a real variable. 1.1 THE FIELD, POSITIVITY, AND COMPLETENESS AXIOMS We assume as given the set R of real numbers such that for each pair of real numbers a and b, there are defined real numbers a + b and ab called the sum and product, respectively, of a and b for which the following Field Axioms, Positivity Axioms, and Completeness Axiom are satisfied. The field axioms Commutativity of Addition: For all real numbers a and b, a + b = b + a. Associativity of Addition: For all real numbers a, b,andc, a + b + c = a + b + c. The Additive Identity: There is a real number, denoted by 0, such that 0 + a = a + 0 = a for all real numbers a. 7 8 Chapter 1 The Real Numbers: Sets, Sequences, and Functions The Additive Inverse: For each real number a, there is a real number b such that a + b = 0. Commutativity of Multiplication: For all real numbers a and b, ab = ba. Associativity of Multiplication: For all real numbers a, b,andc, abc = abc. The Multiplicative Identity: There is a real number, denoted by 1, such that 1a = a1 = a for all real numbers a. The Multiplicative Inverse: For each real number a Z 0, there is a real number b such that ab = 1. The Distributive Property: For all real numbers a, b,andc, ab + c = ab + ac. The Nontriviality Assumption: 1 Z 0. Any set that satisfies these axioms is called a field. It follows from the commutativity of addition that the additive identity, 0, is unique, and we infer from the commutativity of multiplication that the multiplicative unit, 1, also is unique. The additive inverse and multiplicative inverse also are unique. We denote the additive inverse of a by −a and, if aZ0, its multiplicative inverse by a−1 or 1/a. If we have a field, we can perform all the operations of elementary algebra, including the solution of simultaneous linear equations. We use the various consequences of these axioms without explicit mention.1 The positivity axioms In the real numbers there is a natural notion of order: greater than, less than, and so on. A convenient way to codify these properties is by specifying axioms satisfied by the set of positive numbers. There is a set of real numbers, denoted by P, called the set of positive numbers. It has the following two properties: P1 If a and b are positive, then ab and a + b are also positive. P2 For a real number a, exactly one of the following three alternatives is true: a is positive, −a is positive, a = 0. 1A systematic development of the consequences of the Field Axioms may be found in the first chapter of the classic book A Survey of Modern Algebra by Garrett Birkhoff and Saunders MacLane [BM97]. Section 1.1 The Field, Positivity, and Completeness Axioms 9 The Positivity Axioms lead in a natural way to an ordering of the real numbers: for real numbers a and b, we define a>bto mean that a − b is positive, and a ≥ b to mean that a>b or a = b. We then define a<bto mean that b>a,anda ≤ b to mean that b ≥ a. Using the Field Axioms and the Positivity Axioms, it is possible to formally establish the familiar properties of inequalities (see Problem 2). Given real numbers a and b for which a<b, we define a, b ={x | a<x<b}, and say a point in a, b lies between a and b. We call a nonempty set I of real numbers an interval provided for any two points in I, all the points that lie between these points also belong to I. Of course, the set a, b is an interval, as are the following sets: [a, b] = x a ≤ x ≤ b ; [a, b = x a ≤ x<b ; a, b] = x a<x≤ b . (1) The completeness axiom A nonempty set E of real numbers is said to be bounded above provided there is a real number b such that x ≤ b for all x ∈ E : the number b is called an upper bound for E. Similarly, we define what it means for a set to be bounded below and for a number to be a lower bound for a set. A set that is bounded above need not have a largest member. But the next axiom asserts that it does have a smallest upper bound. The Completeness Axiom Let E be a nonempty set of real numbers that is bounded above. Then among the set of upper bounds for E there is a smallest, or least, upper bound. For a nonempty set E of real numbers that is bounded above, the least upper bound of E, the existence of which is asserted by the Completeness Axiom, will be denoted by l.u.b. S. The least upper bound of E is usually called the supremum of E and denoted by sup S. It follows from the Completeness Axiom that every nonempty set E of real numbers that is bounded below has a greatest lower bound; it is denoted by g.l.b. E and usually called the infimum of E and denoted by inf E. A nonempty set of real numbers is said to be bounded provided it is both bounded below and bounded above. The triangle inequality We define the absolute value of a real number x, |x|, to be x if x ≥ 0 and to be −x if x<0. The following inequality, called the Triangle Inequality, is fundamental in mathematical analysis: for any pair of real numbers a and b, |a + b|≤|a|+|b|. The extended real numbers It is convenient to introduce the symbols ∞ and −∞ and write −∞ <x<∞ for all real numbers x. We call the set R ∪ ; ∞ the extended real numbers. If a nonempty set E of real numbers is not bounded above we define its supremum to be ∞. It is also convenient to define −∞ to be the supremum of the empty-set. Therefore every set of real numbers has a supremum that belongs to the extended real-numbers. Similarly, we can extend the concept of infimum so every set of real numbers has an infimum that belongs to the extended real numbers. We will define limits of sequences of real numbers and it is convenient to 10 Chapter 1 The Real Numbers: Sets, Sequences, and Functions allow limits that are extended real numbers. Many properties of sequences of real numbers that converge to real numbers, such as the limit of the sum is the sum of the limits and the limit of the product is the product of the limit, continue to hold for limits that are ;∞, provided we make the following extension of the meaning of sum and product: ∞+∞=∞, −∞−∞ = −∞ and, for each real number x, x+∞ = ∞ and x−∞=−∞;if x>0,x·∞ = ∞ and x·−∞ =−∞while if x<0,x·∞ = −∞and x·−∞ =∞. We define −∞, ∞ = R. For a, b ∈ R, we define a, ∞ = x ∈ R a<x ,−∞,b= x ∈ R x<b and [a, ∞ = x ∈ R a ≤ x ,−∞,b] = x ∈ R x ≤ b . Sets of the above form are unbounded intervals. We leave it as an exercise to infer from the completeness of R that all unbounded intervals are of the above type and that all bounded intervals are of the form listed in (1) together with intervals of the form a, b. PROBLEMS 1. For a Z 0andb Z 0, show that ab−1 = a−1b−1. 2. Verify the following: (i) For each real number a Z 0, a2 > 0. In particular, 1 > 0since1Z 0and1= 12. (ii) For each positive number a, its multiplicative inverse a−1 also is positive. (iii) If a>b,then ac > bc if c>0andac < bc if c<0. 3. For a nonempty set of real numbers E, show that inf E = sup E if and only if E consists of a single point. 4. Let a and b be real numbers. (i) Show that if ab = 0, then a = 0orb = 0. (ii) Verify that a2 − b2 = a − ba + b and conclude from part (i) that if a2 = b2, then a = b or a =−b. (iii) Let c be a positive real number. Define E ={x ∈ R | x2 <c.} Verify that E is nonempty = 2 = and bounded above. Define x0 sup E. Show that x0 √c. Use part (ii) to show that there is a unique x>0forwhichx2 = c.
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