CHAPTER 2InequalitiesIn this section we add the axioms describe the behavior of inequal-ities (theorder axioms) to the list of axioms begun in Chapter 1.A thorough mastery of this section is essential as analysis is based oninequalities.Before describing the additional axioms, however, let us first ask,“What, exactly, is an inequality?” Addition is a binary operation; ittakes two numbersaandband produces a third,a+b. Less thanis abinary relation: it takes two numbersaandband produceseither the value ‘true’ or ‘false’. Mathematically, we would say that<is a function whose domain is the set of all pairs of real numbersand whose range is the set{true, false}. Thus 2<3 produces ‘true’and 3<2 produces ‘false’. If we writea < bwithout explanation, weare asserting thata < bis true.Order AxiomsI1:(Trichotomy) For real numbersaandb, one and only one,of the following statements must hold:(1)a < b(2)b < a(3)a=b.I2:(Transitivity) Ifa < bandb < c, thena < c.I3:(Additivity) Ifa < bandcis any real number, thena+c <b+c.I4:(Multiplicativity) Ifa < bandc >0, thenac < bc.Important!Throughout this text, in our proofs, we will typicallyonly give reasons for material from the current chapter.Hence, indoing proofs with inequalities, we will typically not explicitly indicate17

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182. INEQUALITIESthe use of field axioms such as associativity, commutativity, etc. Sim-ilarly, in Chapter 3, we will not typically indicate the use of the orderaxioms in our proofs.We definea > bto meanb < a.The statementa≤bis acompound statement.It is true if eithera < bor ifa=b.Thus2≤3 and 3≤3 are both true statements. The symbol ‘≥’ is definedsimilarly.There are many rules for studying inequalities which are derivablefrom the axioms. The reader will be asked to prove many of them inthe exercises.These are not axioms.Theorem1.Leta,b,c, anddbe real numbers. ThenE1:(Inequalities add) Ifa < bandc < d, thena+c < b+d.E2:(Positive inequalities multiply) If 0< a < band 0< c < d,then 0< ac < bd.E3:(Multiplication by negatives reverses inequalities) Ifa < bandc <0, thenac > bc.E4:(Inversion reverses inequalities) If 0< a < b, then1a>1b>0.E5:(The product to two negatives is positive) Ifa <0 andb <0 thenab >0.E6:Ifab >0 then either bothaandbare positive or they areboth negative.E7:For alla,a2≥0.E8:Ifa∈N,a >0.(Recall thatNis the set of naturalnumbers.)Remark:In the following example we use interval notation familiarfrom calculus. Thus, ifaandbare real numbers witha < b, then(a, b) is the set ofxsuch thata < x < b. Use of a bracket instead ofa parenthesis indicates that the corresponding end point is included.Hence, for example, [a, b) is the set ofxsuch thata≤x < b. Use of∞as a right end point, or−∞as a left endpoint, indicates that theinterval has no endpoint on that side. Note, however, that∞is NOTA NUMBER! Thus, for example, there is no interval “(−1,∞].”In general, a set is just a collection of objects. The objects in theset are theelementsof the set. We write “x∈A” as shorthand for “xis an element ofA.” Hence,x∈(2,