Studying Rudin's Principles of Mathematical Analysis Through ...

1.9. CONCEPTS FROM THE 2ED 11
1.8.5 Step 5
Having proved that the addition defined in Step 4 satisfies Axioms (A) of Definition 1.12, it follows that
Proposition 1.14 is valid in R. Prove one of the requirements of Definition 1.17: If α, β, γ in R and
β < γ then, α + β < α + γ and that α > 0* if and only if −α < 0*.
1.8.6 Step 6
Multiplication is a little more bothersome than addition in the present context, since products of negative
rationals are positive. For this reason we confine ourselves first to R + , the set of all α ∈ R with α > 0*.
If α, β ∈ R + , define αβ to be the set of all p such that p ≤ rs for some choice of r ∈ α, s ∈ β, r >
0, s > 0. Define 1* to be the set of all q < 1.
Then, prove that the axioms (M) and (D) of Definition 1.12 hold, with R + in place of F and with 1*
in the role of 1.
1.8.7 Step 7
Define αβ, and prove the distributive law α(β + γ) = αβ + αγ. Thus completing the proof that R is an
ordered field with least-upper-bound property.
1.8.8 Step 8
Associate with each r ∈ Q the set r* which consists of all p ∈ Q such that p < r. It is clear that each r*
is a cut; that is, r* ∈ R. Prove that these cuts satisfy the following relations:
(a) r* + s* = (r + s)*;
(b) r*s* = (rs)*;
(c) r* < s* if and only if r < s
1.8.9 Step 9
1.9 Concepts From the 2ed
Some these definitions and theorems are covered in the appendix in the 3rd edition (see above).
1.9.1 Definition
What is a cut?
1.9.2 Theorem
If p ∈ α, and q ∉ α, prove that p < q.
1.9.3 Theorem
Let r be rational. Let α be the set consisting of all rational p such that p < r. Prove that α is a cut, and
r is the smallest upper number of α.
1.9.4 Definition
What is a rational cut?
1.9.5 Definition
Let α and β be cuts. Define α = β and α ≠ β.