• Rudin-《数学分析原理》1.1


    Chapter 1 Real and Complex Number Systems

    INTRODUCTION

    A satisfactory discussion of the main concepts of analysis (such as convergence, continuity, differentiation, and integration) must be based on an accurately defined number concept. We shall not, however, enter into any discussion of the axioms that govern the arithmetic of the integers, but assume familiarity with the rational numbers (i.e., the numbers of the form 1.1 Introduction_1, where 1.1 Introduction_2 and 1.1 Introduction_3 are integers and 1.1 Introduction_4).

            The rational number system is inadequate for many purposes, both as a field and as an ordered set. (These terms will be defined in Secs. 1.6 and 1.12.) For instance, there is no rational 1.1 Introduction_5 such that 1.1 Introduction_6. (We shall prove this presently.) This leads to the introduction of so-called "irrational numbers" which are often written as infinite decimal expansions and are considered to be "approximated" by the corresponding finite decimals. Thus the sequence

    1.1 Introduction_7

    "tends to 1.1 Introduction_8." But unless the irrational number 1.1 Introduction_9 has been clearly defined, the question must arise: Just what is it that this sequence "tends to"?

            This sort of question can be answered as soon as the so-called "real number system" is constructed.

    1.1 Example

             We now show that the equation

    (1)

    1.1 Introduction_10

    is not satisfied by any rational 1.1 Introduction_11. If there were such a 1.1 Introduction_12, we could write 1.1 Introduction_13 where 1.1 Introduction_14 and 1.1 Introduction_15 are integers that are not both even. Let us assume this is done. Then (1) implies

    (2)

    1.1 Introduction_16

    This shows that 1.1 Introduction_17 is even. Hence 1.1 Introduction_18 is even (if 1.1 Introduction_19 were odd, 1.1 Introduction_20 would be odd), and so 1.1 Introduction_21 is divisible by 1.1 Introduction_22. It follows that the right side of (2) is divisible by 1.1 Introduction_23, so that 1.1 Introduction_24 is even, which implies that 1.1 Introduction_25 is even.

            The assumption that (1) holds thus leads to the conclusion that both 1.1 Introduction_26 and 1.1 Introduction_27 are even, contrary to our choice of 1.1 Introduction_28 and 1.1 Introduction_29. Hence (1) is impossible for rational 1.1 Introduction_30.

            We now examine this situation a little more closely. Let 1.1 Introduction_31 be the set of all positive rationals 1.1 Introduction_32 such that 1.1 Introduction_33 and let 1.1 Introduction_34 consist of all positive rationals 1.1 Introduction_35 such that 1.1 Introduction_36. We shall show that 1.1 Introduction_37 contains no largest number and 1.1 Introduction_38 contains no smallest.

            More explicitly, for every 1.1 Introduction_39 in 1.1 Introduction_40 we can find a rational 1.1 Introduction_41 in 1.1 Introduction_42 such that 1.1 Introduction_43, and for every 1.1 Introduction_44 in 1.1 Introduction_45 we can find a rational 1.1 Introduction_46 in 1.1 Introduction_47 such that 1.1 Introduction_48.

            To do this, we associate with each rational 1.1 Introduction_49 the number

    (3)

    1.1 Introduction_50

    Then

    (4)

    1.1 Introduction_51

            If 1.1 Introduction_52 is in 1.1 Introduction_53 then 1.1 Introduction_54, (3) shows that 1.1 Introduction_55, and (4) shows that 1.1 Introduction_56. Thus 1.1 Introduction_57 is in 1.1 Introduction_58.         

            If 1.1 Introduction_59 is in 1.1 Introduction_60 then 1.1 Introduction_61, (3) shows that 1.1 Introduction_62, and (4) shows that 1.1 Introduction_63. Thus 1.1 Introduction_64 is in 1.1 Introduction_65.

    1.2 Remark

            The purpose of the above discussion has been to show that the rational number system has certain gaps, in spite of the fact that between any two rationals there is another: If 1.1 Introduction_66 then 1.1 Introduction_67. The real number system fills these gaps. This is the principal reason for the fundamental role which it plays in analysis.

            In order to elucidate its structure, as well as that of the complex numbers, we start with a brief discussion of the general concepts of ordered set and field.

            Here is some of the standard set-theoretic terminology that will be used throughout this book.

    1.3 Definitions

            If 1.1 Introduction_68 is any set (whose elements may be numbers or any other objects), we write  1.1 Introduction_69 to indicate that 1.1 Introduction_70 is a member (or an element) of 1.1 Introduction_71. If 1.1 Introduction_72 is not a member of 1.1 Introduction_73, we write: 1.1 Introduction_74.

            The set which contains no element will be called the empty set. If a set has at least one element, it is called nonempty.

            If 1.1 Introduction_75 and 1.1 Introduction_76 are sets, and if every element of 1.1 Introduction_77 is an element of 1.1 Introduction_78, we say that 1.1 Introduction_79 is a subset of 1.1 Introduction_80, and write 1.1 Introduction_81, or 1.1 Introduction_82. If, in addition, there is an element of 1.1 Introduction_83 which is not in 1.1 Introduction_84, then 1.1 Introduction_85 is said to be a proper subset of 1.1 Introduction_86. Note that 1.1 Introduction_87 for every set 1.1 Introduction_88.

            If 1.1 Introduction_89 and 1.1 Introduction_90 , we write 1.1 Introduction_91. Otherwise 1.1 Introduction_92.

    1.4 Definition

    Throughout Chap. 1, the set of all rational numbers will be denoted by 1.1 Introduction_93.

    See you next time---! 88

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