Math 320 { September 20, 2020 7 The completeness axiom Definition 7.1. Let S ⊆R. If there exists a real number M, such that M ≥s for all s2S, then M is called an upper bound for S, and we say that S is bounded above. If there exists a real number m, such that m≤s for all s2S, then m is called a lower bound for S, and we say that S is bounded below. A set is called bounded, if it is bounded above and below. If M 2S is an upper bound for S, then M is called the maximum of S, M =maxS, and if m2S is a lower bound for S, then m is called the minimum of S, m=minS. Definition 7.2. Let S ⊆R be nonempty. If S is bounded above, then the least upper bound is called the supremum of S, denoted by supS. If S is bounded below, then the greatest lower bound is called its infimum, denoted by infS. Thus, M = supS is equivalent to the conditions (similar conditions hold for the infimum with reversed inequalities) (a) M ≥s for all s2S (b) If M0 ≥s for all s2S, then M0 ≥M. The contrapositive of the last condition is: (b0) If M0 <M, then there exists s2S, such that s≥M0. Axiom 7.3 (The completeness axiom). Every nonempty subset of R that is bounded above has a least upper bound. That is, supS exists, and supS 2R. The existence of the infimum for sets bounded below follows from the completeness axiom by considering the set −S =f−s:s2Sg, and observing that infS =−sup(−S). A consequence of the completeness axiom is the following property. Theorem 7.4 (Archimedean property). The set N of natural numbers is unbounded above in R. We have the following equivalent statements to the Archimedean property. Theorem 7.5. The Archimedean property is equivalent to: (a) 8z 2R;9n2N, such that n>z (b) 8x>0 and y2R;9n2N such that nx>y 1 (c) 8x>0;9n2N such that 0< n <x. Another consequence of the completeness axiom is the following. Theorem 7.6 (Nested interval property). Every sequence of nested closed intervals has a nonempty intersection. T1 That is, if In =[an;bn]=fx2R:an ≤x≤bng for every n2N, and I1 ⊇I2 ⊇I3 ⊇I4 ⊇:::, then n=1In =6 ;. Using the completeness one can also prove the following theorem. Theorem 7.7. If p2N is a prime number, then there exists a real number x, such that x2 =p. Moreover, x2 = Q. Using the equivalent statements to the Archimedean property, one can show that both the rational and irrational numbers are dense in R. Theorem 7.8. If x;y 2 R, with x < y, then there exists a rational number r 2 Q, and an irrational number w2RnQ, such that x<r<y, and x<w<y..