By Derek G. Ball and C. Plumpton (Auth.)

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DEFINITION. If a set of numbers is such that any of its bounded subsets has a least upper bound, the set is called complete. 2 proves that the set of real numbers is complete. 4. 28 AN INTRODUCTION TO REAL ANALYSIS Postulates for the real numbers. We may summarize the properties of the real numbers as follows : (1) Addition and multiplication are commutative and associative, and multiplication is distributive over addition. Subtraction and division (other than by zero) are always possible. We summarize all these properties by saying that the real numbers constitute a field.

Consider the set B = {—w, —x, —y9 . . } consisting of all those real numbers which are negatives of members of A. Then B is bounded above and so has a least upper bound, say M. Then — M will be the greatest lower bound for A. D COROLLARY. DEFINITION. If a set of numbers is such that any of its bounded subsets has a least upper bound, the set is called complete. 2 proves that the set of real numbers is complete. 4. 28 AN INTRODUCTION TO REAL ANALYSIS Postulates for the real numbers. We may summarize the properties of the real numbers as follows : (1) Addition and multiplication are commutative and associative, and multiplication is distributive over addition.

1, V2, V3, 2, V5, V6, V7, V8, 3, . . , 1, — 1, 1, — 1, 1, — 1, 1, . . , are all sequences. Since we talk about the first term, second term, third term of a sequence and so on, it is natural to make the following definition. DEFINITION. A sequence is a function whose domain is the set of natural numbers and whose range is some subset of the real numbers. The sequences above are defined by the functions/^)=H, f(n) = \Jn f(n) = 1/«, f(n) = ( — l)n+1 respectively. It is common practice to write the nth term of a sequence an instead of/(«); in this notation 36 37 SEQUENCES the definition of the last sequence would read an = ( —1)Λ+1.