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. 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 0 so that x ≤ M for all x ∈ S), then l.u.b. However, one can prove the Axiom of Completeness if one defines the real numbers as infinite decimals.12 Math 299 Lecture 33: Real Numbers and the Completeness Axiom De nitions: Let Sbe a nonempty subset of R, i.e. (3)If 9M2R such that x Mforall x2S, then Mis called an upper bound of Sand the set Sis bounded above. Equivalently, R is complete. IV.

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