4 1. THE CONSTRUCTION OF REAL AND COMPLEX NUMBERS Exercise 1.1.12. Let n be a positive integer that is not a perfect square. Let A = {x ∈ Q | x2 n}. Show that A is bounded in Q but has neither a greatest lower bound nor a least upper bound in Q. Conclude that √ n exists in R, that is, there exists a real number a such that a2 = n. We have observed that the rational numbers are contained in R. A real number is irrational if it is not in Q. Fact 1.1.13. We can conclude from Exercise 1.1.12 that if n is a positive integer that is not a perfect square, then √ n exists in R and is irrational. Exercise 1.1.14. Suppose that A and B are bounded sets in R. Prove or disprove the following: (i) lub(A ∪ B) = max{lub A, lub B}. (ii) If A + B = {a + b | a ∈ A, b ∈ B}, then lub(A + B) = lub A + lub B. (iii) If the elements of A and B are positive and A·B = {ab | a ∈ A, b ∈ B}, then lub(A · B) = (lub A)(lub B). (iv) Formulate the analogous problems for the greatest lower bound. 1.2. Consequences of the Least Upper Bound Property We now present some facts which follow from the least upper bound property and the properties of the integers. The first is the Archimedean Property of the real numbers. Theorem 1.2.1 (Archimedean property of R). If a and b are positive real numbers, then there exists a natural number n such that na b. Proof. If a b, take n = 1. If a = b, take n = 2. If a b, consider the set S = {na | n ∈ N}. The set S = ∅ since a ∈ S. Suppose S is bounded above by b. Let L = lub S. Then, since a 0, there exists an element n0a ∈ S such that L−a n0a. But then L (n0+1)a, which is a contradiction. Corollary 1.2.2. If ε is a positive real number, then there exists a natural number n such that 1/n ε. Definition 1.2.3. Let F be an ordered field. From Exercise 1.0.1, we know that Z ⊆ F and by Exercise 1.0.2 we know Q ⊆ F . We say that F is an Archimedean ordered field if for every x ∈ F there exists N ∈ Z such that x N. The fields Q and R are Archimedean ordered fields. Exercise 1.2.4. Let F be an Archimedean ordered field. Show that F is order isomorphic to a subfield of R. Next, we show that every real number lies between two successive inte- gers. Theorem 1.2.5. If a is a real number, then there exists an integer N such that N − 1 ≤ a N.

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