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 s∈S, 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 s∈S, 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 ∈S is an upper bound for S, then M is called the maximum of S, M =maxS, and if m∈S 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 s∈S (b) If M0 ≥s for all s∈S, then M0 ≥M. The contrapositive of the last condition is: (b0) If 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 ∈R. The existence of the infimum for sets bounded below follows from the completeness axiom by considering the set −S ={−s:s∈S}, 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) ∀z ∈R,∃n∈N, such that n>z (b) ∀x>0 and y∈R,∃n∈N such that nx>y 1 (c) ∀x>0,∃n∈N such that 0< n

Theorem 7.7. If p∈N is a prime number, then there exists a real number x, such that x2 =p. Moreover, x∈ / 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 ∈ R, with x < y, then there exists a rational number r ∈ Q, and an irrational number w∈R\Q, such that x