# Real-nos - PART I THE REAL NUMBERS This material assumes...

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Unformatted text preview: PART I. THE REAL NUMBERS This material assumes that you are already familiar with the real number system and the represen- tation of the real numbers as points on the real line. I.1. THE NATURAL NUMBERS AND INDUCTION Let N denote the set of natural numbers (positive integers). Axiom: If S is a nonempty subset of N , then S has a least element. That is, there is an element m ∈ S such that m ≤ n for all n ∈ S . Note: A set which has the property that each non-empty subset has a least element is said to be well-ordered . Thus, the axiom tells us that the natural numbers are well-ordered. Mathematical Induction. Let S be a subset of N . If S has the following properties: 1. 1 ∈ S , and 2. k ∈ S implies k + 1 ∈ S , then S = N . Proof: Suppose S = N . Let T = N- S . Then T = ∅ . Let m be the least element in T . Then m- 1 / ∈ T . Therefore, m- 1 ∈ S which implies that ( m- 1) + 1 = m ∈ S , a contradiction. Corollary: Let S be a subset of N such that 1. m ∈ S . 2. If k ≥ m ∈ S , then k + 1 ∈ S . Then, S = { n ∈ N : n ≥ m } . Example Prove that 1 + 2 + 2 2 + 2 3 + ··· + 2 n- 1 = 2 n- 1 for all n ∈ N . SOLUTION Let S be the set of integers for which the statement is true. Since 2 = 1 = 2 1- 1, 1 ∈ S. Assume that the positive integer k ∈ S. Then 2 + 2 1 + ··· + 2 k- 1 + 2 k = ( 2 + 2 1 + ··· + 2 k- 1 ) + 2 k = 2 k- 1 + 2 k = 2 · 2 k- 1 = 2 k +1- 1 . Thus, k + 1 ∈ S. 1 We have shown that 1 ∈ S and that k ∈ S implies k + 1 ∈ S . It follows that S contains all the positive integers. Exercises 1.1 1. Prove that 1 + 2 + 3 + ··· + n = n ( n + 1) 2 for all n ∈ N . 2. Prove that 1 2 + 2 2 + 3 2 + ··· + n 2 = n ( n + 1)(2 n + 1) 6 for all n ∈ N . 3. Let r be a real number r = 1. Prove that 1 + r + r 2 + r 3 + ··· + r n = 1- r n +1 1- r . for all n ∈ N 4. Prove that 1 + 2 n ≤ 3 n for all n ∈ N . 5. Prove that 1 √ 1 + 1 √ 2 + 1 √ 3 + ··· + 1 √ n ≥ √ n for all n ∈ N . 6. Prove that 1- 1 2 2 1- 1 3 2 ··· 1- 1 n 2 = n + 1 2 n for all n ≥ 2. 7. True or False: If S is a non-empty subset of N , then there exists an element m ∈ S such that m ≥ k for all k ∈ S . I.2. ORDERED FIELDS Let R denote the set of real numbers. The set R , together with the operations of addition (+) and multiplication ( · ), satisfies the following axioms: Addition: A1. For all x, y ∈ R , x + y ∈ R (addition is a closed operation). A2. For all x, y ∈ R , x + y = y + x (addition is commutative) A3. For all x, y, z ∈ R , x + ( y + z ) = ( x + y ) + z (addition is associative). A4. There is a unique number 0 such that x +0 = 0+ x for all x ∈ R . (0 is the additive identity .) A5. For each x ∈ R , there is a unique number- x ∈ R such that x + (- x ) = 0. (- x is the additive inverse of x .) Multiplication: M1. For all x, y ∈ R , x · y ∈ R (multiplication is a closed operation)....
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Real-nos - PART I THE REAL NUMBERS This material assumes...

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