Chapter 15

Construction of Real NumbersThe purpose of this chapter is to give a construction of the real numbers from the rationals.This was first done by Dedekind in 1858 although not published till 1872. He used a totallydifferent approach than what is used here. Dedekind’s construction was based on an earlyidea of Eudoxus who lived around 350 B.C. in which points on the real line divided therationals above and below them. Dedekind made this cut into what was meant by a realnumber. See Rudin [24] for a description of this approach. I am using equivalence classesof Cauchy sequences rather than Dedekind cuts because this approach applies to metricspaces and Dedekind cuts do not. This idea of constructing the real numbers from suchequivalence classes is due to Cantor, also published in 1872. Hewitt and Stromberg [17]also use this approach. Hobson [18] has a description of both of these methods.

Why do this? Why not continue believing that a real number is a point on a numberline and regard its existence as geometrically determined? This is essentially what was donetill Dedekind and, as mentioned, seems to have been part of the belief system of Greeksthousands of years earlier. I think the reason that such a construction is needed is algebra.Is there a way to carry the algebraic notions and order axioms of the rational numbers to thereal numbers, defined geometrically as points on the number line? Till Dedekind, this wassimply assumed, as it has been in this book till now. Also, it is desirable to remove the lastvestiges of geometry from analysis; hence this construction. I think this is why Dedekindand Cantor’s work was very important.

Definition 15.0.1 Let R denote the set of Cauchy sequences of rational numbers.If {xn}∞

n=1 is such a Cauchy sequence, this will be denoted by x for the sake of simplicity ofnotation. A Cauchy sequence x will be called a null sequence if limn→∞ xn = 0. Denote thecollection of such null Cauchy sequences as N.Then define x∼ y if and only if x−y ∈ N.Here x−y signifies the Cauchy sequence {xn− yn}∞

n=1. Also, for simplicity of notation, letQ denote the collection of Cauchy sequences equivalent to some constant Cauchy sequence{an}∞

n=1 for an = a ∈Q. Thus Q⊆ R.

Notice that whether x−y ∈ N is determined completely by the tail of x−y, the termsof the sequence larger than some number. Then the following proposition is very easy toestablish and is left to the reader.

Proposition 15.0.2 ∼ is an equivalence relation on R.

Definition 15.0.3 Define R as the set of equivalence classes of R. For [x] , [y] , [z]∈R, define [x] [y]≡ [xy] where xy is the sequence {xnyn}∞

n=1 . Also define [x]+ [y]≡ [x+y] .

This leads to the following theorem. Note that this is enlarging the field Q obtaining alarger field R. Enlarging fields is done frequently in algebra using the machinery of fieldextensions. It also uses equivalence classes. However, this is very different, resulting in anenlargement of Q which essentially goes all the way at once. It emphasizes completenessand order rather than inclusion of roots of various polynomials.

Lemma 15.0.4 If x /∈ N, then there is δ > 0 and N such that |xk|> δ for all k ≥ N.

Proof: If the conclusion does not hold, then for each δ > 0, there exist infinitely manyk such that |xk| ≤ δ . Thus there is a subsequence which converges to 0. By Theorem 4.5.4,x ∈ N after all.

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