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Construction of Real Numbers

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Construction of Real Numbers Appendix A Construction of Real Numbers The purpose of this appendix is to give a construction of the field R of real numbers from the field Q of rational numbers which, we assume, is known to the reader. Let us point out that we did not give the axiomatic construction of the set N of natural numbers from which one can first construct the set Z of integers and, subsequently, the set Q of rationals. These constructions may be found in most textbooks on abstract algebra, e.g., A Survey of Modern Algebra, by Birkhoff and MacLane [BM77]. Most authors use the so-called Dedekind Cuts to construct the set of real numbers from that of rational numbers. Since, however, the reader is now familiar with sequences and series, it is more natural to use Georg Cantor’s method of construction, which is based on Cauchy sequences of rational numbers, and can be extended to more abstract situations. This abstraction, which is referred to as the completion of a metric space, was discussed in Chap. 5. We begin our discussion by introducing some notation and definitions. Notation. We recall that the set of rational numbers is denoted by Q and the set of positive rationals by QC Dfr 2 Q W r>0g: Also, the set of all sequences of rational numbers, i.e., the set of all functions from N to Q; is denoted by QN: Next, we define the Cauchy sequences of rational numbers. Although the definition of Cauchy sequences was given earlier, since we have not yet constructed the set of real numbers, we must insist that the ">0in our definition take on rational values only. Definition A.1 (Cauchy Sequences in Q). A sequence x 2 QN is called a Cauchy sequence if the following holds: C .8" 2 Q /.9N 2 N/.m; n N )jxm xnj <"/: The set of all Cauchy sequences in Q will be denoted by C: © Springer Science+Business Media New York 2014 655 H.H. Sohrab, Basic Real Analysis, DOI 10.1007/978-1-4939-1841-6 656 A Construction of Real Numbers Next, we define null sequences in Q: Again, this was defined earlier, but we must be careful to use only rational ">0. Definition A.2 (Null Sequences in Q). A sequence x 2 QN is called a null sequence if the following holds: C .8" 2 Q /.9N 2 N/.n N )jxnj <"/: The set of all null sequences in Q will be denoted by N : Remark. Note that N is precisely the set of all rational sequences that converge to zero and that we obviously have N C: (Why?) As we have seen, the set Q of rationals is dense in the set R of real numbers, which we introduced axiomatically. It follows that each real number is the limit of a(not unique) sequence .xn/ of rational numbers. It is tempting, therefore, to take such a sequence .xn/ as the definition of the real number : The nonuniqueness of .xn/ poses a problem, however, for two such sequences in fact represent the same : This motivates the following definition. Definition A.3 (Equivalent Cauchy Sequences). We say that two Cauchy sequences x;y 2 C are equivalent and write x y; if and only if x y 2 N : Exercise A.1. Show that the relation is indeed an equivalence relation on the set C: Notation. For each sequence x 2 C; its equivalence class is denoted by Œx and, we recall, is defined by Œx Dfy 2 C W y xg: The set of all equivalence classes of elements of C is denoted by C=N : Definition A.4 (Real Number). The set R of real numbers is defined to be R WD C=N : Thus is a real number if D Œx for some x 2 C: The sequence x 2 C is then called a representative of : Clearly, if x and y both represent ; then x y 2 N . Exercise A.2. 1. Show that a Cauchy sequence in Q is bounded. 2. Show that C is closed under addition and multiplication; i.e., 8x; y 2 C; we have x C y; xy 2 C: 3. Show that N is an ideal in CI i.e., it is closed under addition and satisfies the stronger condition that 8x 2 N and 8y 2 C we have xy 2 N : Hints: For the addition, use an "=2-argument. For the multiplication, use the inequalities jxmym xnynjÄjymjjxm xnjCjxnjjym ynjÄBjxm xnjCAjym ynj; for some constants A; B 2 QC; where the second inequality follows from part (1). Definition A.5 (Addition, Subtraction, Multiplication). Let D Œx and D Œy be any real numbers. We define C ; ; ; and (or ) as follows: A Construction of Real Numbers 657 1. C WD Œx C y; 2. WD Œy; 3. WD C ./ D Œx y; and 4. WD Œxy: Exercise A.3. Show that the definitions of C and are independent of the representatives x and y of and ; respectively. In other words, show that, if x x0 and y y0; then we have x C y x0 C y0 and xy x0y0: Hint: You will need arguments similar to those needed in Exercise A.2. Proposition A.1 (Ring Properties of R). The set R of real numbers is a commuta- tive ring with identity. In other words, for all real numbers ; ; and ; we have 1. C D C ; 2. C / C D C . C /; 3. 9 0 2 R with 0 C D ; 4. 9 2 R with C ./ D 0; 5. D ; 6. ./ D ./; 7. 9 1 2 R;1¤ 0; with 1 D ; and 8. .C / D C : Proof. The proofs of these properties are straightforward. For example, to prove (2), note that if D Œx; D Œy; and D Œz; then . C / C D Œ.x C y/ C z; while C.C/ D ŒxC.yCz/: Since we obviously have .xCy/Cz D xC.yCz/ in C; (2) follows. Note that the additive identity (“0” in (3)) is in fact 0 D Œ.0; 0; 0; : : :/ 2 N and that the multiplicative identity (“1” in (7)) is 1 D Œ.1; 1; 1; : : :/: Also, 1 ¤ 0 is obvious, because the sequences .0;0;0;:::/and .1;1;1;:::/are not equivalent. ut Proposition A.2. Let W Q ! R be defined by .r/ D Œ.r; r; r; : : :/: Then is an injective “ring homomorphism.” In other words, is a one-to-one map satisfying .r C s/ D .r/ C .s/; .rs/ D .r/.s/; .0/ D 0; and .1/ D 1; 8 r; s 2 Q: Exercise A.4. Prove Proposition A.2. Remark. By Proposition A.2,themap is a field isomorphism of Q onto its image .Q/ RI i.e., a one-to-one correspondence between Q and .Q/ that preserves all the algebraic properties of Q: Therefore, we henceforth identify the two sets and, by abuse of notation, will write Q D .Q/ R: Based on this identification, the field Q of rational numbers becomes a subfield of the field R of real numbers. Here, by a field we mean a set F together with two operations “+” of addition and “”of multiplication, i.e., two maps CW.x; y/ 7! x Cy and W.x; y/ 7! x y; from FF to F; satisfying the nine (algebraic) axioms (A1 A4;M1 M4;D) stated for real numbers in Sect. 2.1 of Chapter 2. Proposition A.1 only shows that R is a commutative ring with identity. To prove that R is actually a field, the only property we need to check is the existence of reciprocals for nonzero real numbers (cf. Axiom .M4/ at the beginning of Chap. 2). To this end, we shall need the following. 658 A Construction of Real Numbers Proposition A.3. Let be a nonzero element of R: Then, there exists a rational C number r 2 Q and a representative x 2 C of such that either xn r 8n 2 N or xn Är 8n 2 N: Proof. Let y 2 C be a representative of : Since ¤ 0; the sequence .yn/ is not equivalent to .0;0;0;:::/and we have C .9" 2 Q /.8N 2 N/.9n N/.jyn 0j"/: ( ) On the other hand, .yn/ 2 C implies that .9K 2 N/.m; n K )jym ynj <"=2/: ( ) Now, by ( ), we can find k K such that jykj": Changing to ; if necessary, we may assume that yk ": Therefore, using ( ), m K )jym ykj <"=2 ) ym yk jym ykj" "=2 D "=2: Let xn WD "=2 for n<K;and xn D yn for n K: It is then clear that D Œ.xn/ and that, with r WD "=2; we have xn r for all n 2 N. ut Definition A.6 (Positive and Negative Cauchy Sequences). We say that a sequence x 2 C is positive (resp., negative) if it satisfies the first (resp., second) alternative in Proposition A.3. The set of all positive (resp., negative) sequences in C is denoted by CC (resp., C). Remark. It is obvious that the two alternatives in Proposition A.3 are mutually exclusive, i.e., that CC \ C D;: Moreover, the condition in the first (and hence also second) alternative needs only be satisfied ultimately; i.e., it can be replaced by C .9r 2 Q /.9N 2 N/.n N ) xn r/: 0 0 Indeed, one can always replace x by the equivalent sequence x defined by xk WD 0 r 8k<Nand xk WD xk 8k N: Proposition A.4.
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