Avner Friedman - Advanced Calculus

ADVANCED
CALCULUS

PREFACE

FUNCTIONS OF ONE VARIABLE

NUMBERS AND SEQUENCES

1. Tel Aviv, IsraelMay 1971 The positive integers 1,2, 3, are called natural numbers. Since we intend to do things rigorously, we cannot be satisfied with our everyday familiarity with these numbers, and we should try to axiomatize their properties. Let us first write down five statements concerning the natural numbers that we feel should be true: (I) 1 is a natural number. (II) To every natural number n there is associated in a unique way another natural number n called the successor of n. (III) 1 is not a successor of any natural number. (V) Let M be a subset of the natural numbers such that: (i) 1 is in M, and (ii) if a natural number is in M, then its successor also is in M. (V) Let M be a subset of the natural numbers such that: (i) 1 is in M, and (ii) if a natural number is in M, then its successor also is in M.

Then M coincides with the set of all the natural numbers. From now on we consider the statements (I)(V) to be axioms. They are called the Peano axioms. The natural numbers will be the objects occurring in the Peano axioms. Axiom (V) is called the principle of mathematical induction. We denote the successor of 1 by 2, the successor of 2 by 3, and so on.

Note that 2 1. Indeed, if 2 = 1, then 1 is the successor of 1, thus contradicting (III). Note next that 3 2. Indeed, if 3 = 2 then, by (IV), 2 = 1, which is false. In general, one can show that all the numbers obtained by taking the successors of 1 any number of times are all different. The proof of this statement, which we shall not give here, is based on induction, that is, on Axiom (V).

We would like to state Axiom (V) in a form more suitable for application: (V) Let P(n) be a property regarding the natural number n, for any n. Suppose that (i) P(1) is true, and (ii) if P(n) is true, P(n) also is true. Then P(n) is true for all n. If we define M to be the set of all natural numbers for which P(n) is true, then (V) follows from (V). If, on the other hand, we define P(n) to be the property that n belongs to M, then (V) follows from (V). Thus (V) and (V) are equivalent axioms.

The Peano axioms give us objects with which to work. We now proceed to define operations on these objects. There are two operations that we consider: addition (+) and multiplication (). To any given pair of natural numbers each of these operations corresponds another natural number. The precise definition of this correspondence is given in the following theorem. There exist unique operations + and . with the following properties: The proof will not be given here. with the following properties: The proof will not be given here.

We shall often write mn instead of m n. THEOREM 2. The following properties are true for all natural numbers m, n, k: The proof of . We state, without proof, another theorem, known as the trichotomy law: THEOREM 3. Given any natural numbers m and n, one and only one of the following possibilities occurs: (i) m = n. (iii) n = m + y for some natural number y. (iii) n = m + y for some natural number y.

If (ii) holds, we write m > n or n < m, and we say that m is larger or greater than n and that n is smaller or less than m. If either (i) or (ii) holds, we write mn or nm, and say that m is larger or equal to n and that n is less than or equal to m.

1. If n > m, then n + k > m + k. 2. 3. 3.

If n + km