Math 299 Lecture 33: Real Numbers and the Completeness Axiom De nitions: Let Sbe a nonempty subset of R, i.e. The Axiom of Completeness is an important property of real numbers: Axiom of Completeness. S exists. To prove that a property P satisﬁed by the real numbers is not equivalent to completeness, we need to show that there exists an ordered ﬁeld that satisﬁes property P but not the completeness property. THE COMPLETENESS PROPERTY OF R 47 2.4 The Completeness Property of R In this section, we start studying what makes the set of real numbers so special, why the set of real numbers is fundamentally di⁄erent from the set of rational numbers. (See Solved Exercise 10.) (3)If 9M2R such that x Mforall x2S, then Mis called an upper bound of Sand the set Sis bounded above. IV. Every cut determines a real number. Ordinarily, one does not expect to prove this statement, since axioms are basic statements that one does not prove. So it’s very useful to have on hand a number of diﬀerent ordered ﬁelds that are almost the real numbers… Proof. Any interval is a convex set. Exercise \(\PageIndex{2}\) Find two sequences of rational numbers (\(x_n\))and (\(y_n\)) which satisfy properties 1-4 of the NIP and such that there is no rational number \(c\) satisfying the conclusion of the NIP. (2)If x 02Sand x 0 xforall x2S, then x 0 is called theminimumof S. (x 0 = minS.) Every Cauchy sequence of real numbers converges to a real number. n) of real numbers just as we did for rational numbers (now each x n is itself an equivalence class of Cauchy sequences of rational numbers). The completeness axiom asserts the converse. 1 Completeness of R. Recall that the completeness axiom for the real numbers R says that if S ⊂ R is a nonempty set which is bounded above ( i.e there is a positive real number M > 0 so that x ≤ M for all x ∈ S), then l.u.b. Note that we need not state the corresponding axiom for … ˚6= S R (1)If x 02Sand x x 0 forall x2S, then x 0 is called themaximumof S. (x 0 = maxS.) Corollary 1.13. The next problem shows that the completeness property distinguishes the real number system from the rational number system. 50 CHAPTER 4: THE REAL NUMBERS Definition A set S of reai numbers is convex if, whenever Xl and X2 be long to S and Y is a number such thatXl

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