Introduction to Mathematical Analysis I - Second Edition

18 1.4 ORDERED FIELD AXIOMS 1.3.4 Prove the following using induction. (a) 2 n + 1 ≤ 2 n for n ≥ 3 ( n ∈ N ). (b) n 2 ≤ 3 n for all n ∈ N . ( Hint: show first that for all n ∈ N , 2 n ≤ n 2 + 1. This does not require induction.) (c) n 3 ≤ 3 n for all n ∈ N . ( Hint: Check the cases n = 1 and n = 2 directly and then use induction for n ≥ 3.) 1.3.5 Given a real number a 6 = 1, prove that 1 + a + a 2 + · · · + a n = 1 − a n + 1 1 − a for all n ∈ N . 1.3.6 I The Fibonacci sequence is defined by a 1 = a 2 = 1 and a n + 2 = a n + 1 + a n for n ≥ 1 . Prove that a n = 1 √ 5 h 1 + √ 5 2 n − 1 − √ 5 2 n i . 1.3.7 Let a ≥ − 1. Prove by induction that ( 1 + a ) n ≥ 1 + na for all n ∈ N . 1.3.8 B Let a , b ∈ R and n ∈ N . Use Mathematical Induction to prove the binomial theorem ( a + b ) n = n ∑ k = 0 n k a k b n − k , where n k = n ! k ! ( n − k ) ! . 1.4 ORDERED FIELD AXIOMS In this book, we will start from an axiomatic presentation of the real numbers. That is, we will assume that there exists a set, denoted by R , satisfying the ordered field axioms, stated below, together with the completeness axiom, presented in the next section. In this way we identify the basic properties that characterize the real numbers. After listing the ordered field axioms we derive from them additional familiar properties of the real numbers. We conclude the section with the definition of absolute value of a real number and with several results about it that will be used often later in the text. We assume the existence of a set R (the set of real numbers) and two operations + and · (addition and multiplication) assigning to each pair of real numbers x , y , unique real numbers x + y and x · y and satisfying the following properties: (1a) ( x + y )+ z = x +( y + z ) for all x , y , z ∈ R . (1b) x + y = y + x for all x , y ∈ R . (1c) There exists a unique element 0 ∈ R such that x + 0 = x for all x ∈ R .