1.1. LEAST UPPER BOUND PROPERTY AND THE REAL NUMBERS 3 (a) L is an upper bound for A (b) if M is any upper bound for A, then L M. Exercise 1.1.5. Show the least upper bound of a set is unique. We now give a formal definition of the real numbers which provides a working basis for proving theorems. Later in this chapter, starting with the rational numbers as an ordered field we will give a precise construction of the real numbers as an ordered field in which the least upper bound property holds. Definition 1.1.6. The real numbers are an ordered field in which every nonempty subset that is bounded above has a least upper bound and are denoted by the symbol R. We say that the real numbers are an ordered field with the least upper bound property. In many texts, the real numbers are defined as a com- plete ordered field. This is actually a misuse of the word “complete” which is defined in terms of the convergence of Cauchy sequences. This will be discussed later in this chapter. Exercise 1.1.7. Find the least upper bound in R of the set A in Exercise 1.1.3. Definition 1.1.8. Suppose that F and F are ordered integral domains. We say that F and F are order isomorphic if there is a bijection φ : F −→ F such that (a) φ(x + y) = φ(x) + φ(y) for all x, y F (b) φ(xy) = φ(x)φ(y) for all x, y F (c) if x, y F and x y, then φ(x) φ(y) in F . Exercise 1.1.9. Show that any two ordered fields with the least upper bound property are order isomorphic. This exercise proves that if the real numbers exist, they are unique up to order isomorphism. Definition 1.1.10. An ordered field F has the greatest lower bound property if every nonempty subset A of F that is bounded below has a greatest lower bound. That is, there exists an element of F such that: (a) is a lower bound for A (b) if m is any lower bound for A, then m . Exercise 1.1.11. Prove that an ordered field has the least upper bound property iff it has the greatest lower bound property. If L is the least upper bound of a set A, we write L = lub A or L = sup A (sup stands for supremum). If is the greatest lower bound of a set A, we write = glb A or = inf A (inf stands for infimum).
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