1.10. INDEPENDENT PROJECTS 25 Exercise 1.10.3. If α and β are cuts, show that α + β is a cut, and also show that 0 is a cut. Exercise 1.10.4. Show that with this addition (R,+) is an abelian group with 0 as the identity element. We now define an order on R. Definition 1.10.5. If α, β ∈ R, we say that α β if α is a proper subset of β. Exercise 1.10.6. Show that the relation satisfies the following prop- erties: (i) (trichotomy) if α, β ∈ R, then one and only one of the following holds: α β, α = β, or β α (ii) (transitivity) if α, β, γ ∈ R with α β and β γ, then α γ (iii) (additivity) if α, β, γ ∈ R with α β, then α + γ β + γ. It is now possible to define the notions of bounded above, bounded below, bounded, upper bound, least upper bound, lower bound, and greatest lower bound in R just as we did earlier in this chapter. Exercise 1.10.7. Show that the least upper bound property holds in R, that is, if A is a nonempty subset of R which is bounded above, then A has a least upper bound in R. Next, we must define multiplication in R. Definition 1.10.8. If α, β ∈ R with α, β 0, then αβ = {p ∈ Q | there are positive elements r ∈ α and s ∈ β so that p ≤ rs}. The next step is multiplication by 0, which is exactly as it should be, namely for any α ∈ R, we define α0 = 0. Exercise 1.10.9. If α 0 or β 0 or both, replace any negative element by its additive inverse and use the multiplication of positive elements to define multiplication accordingly. For example, if α 0 and β 0, αβ = −[(−α)(β)]. Show that R with addition, multiplication, and order as defined above is an ordered field. Exercise 1.10.10. Put it all together and show that R is an Archimedean ordered field in which the least upper bound property holds. 1.10.2. Infinite Series. An important topic in analysis is the study of infinite series. This theory will be used in the remaining chapters of this book. We assume that the reader has had at least an elementary introduction to infinite series and their convergence properties. In fact, the theory of infinite series actually reduces to the convergence of sequences, which we have covered thoroughly in this chapter. An infinite series is expressed as

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