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

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6 CHAPTER 1. THE REAL AND COMPLEX NUMBER SYSTEMS 1.3.4 Proposition: Implications of the Multiplication Axioms Prove that the axioms for multiplication imply the following statements: (a) If x ≠ 0 and xy = xz, then y = z. (b) If x ≠ 0 and xy = x, then y = 1. (c) If x ≠ 0 and xy = 1, then y = 1/x. (d) If x ≠ 0, then 1/(1/x) = x. 1.3.5 Proposition: Implications of the Multiplication Axioms Prove that the axioms for multiplication imply the following statements, for any x, y, z ∈ F. (a) 0x = 0. (b) If x ≠ 0 and y ≠ 0, then xy ≠ 1. (c) (−x)y = −(xy) = x(−y). (d) (−x)(−y) = xy. 1.3.6 Definition: Ordered Field and Set Define an ordered field, ordered set. 1.3.7 Proposition: Ordered Field Implications Prove that the following statements are true in every ordered field: (a) If x > 0, then −x < 0, and vice versa. (b) If x > 0 and y < z, then xy < xz. (c) If x < 0 and y < z, then xy > xz. (d) If x ≠ 0, then x 2 > 0. In particular, 1 > 0. (e) If 0 < x < y, then 0 < 1/y < 1/x. Hint: Use the field axioms and the definition for an ordered field. 1.4 The Real Field 1.4.1 Theorem: Q ⊂ R and R has the least-upper-bound property Prove that there exists an ordered field R which has the least-upper-bound property; moreover, R contains Q as a subfield.
1.4. THE REAL FIELD 7 Step 1 The members of R will be certain subsets of Q, called cuts. What three properties, by definition, does a cut (any set α) have? The letters p, q, r,... will always denote rational numbers, and α, β, γ, ... will denote cuts. Step 4 If α, β ∈ R, define α + β to be the set of all sums r + s where r ∈ α, s ∈ β. Define 0* to be the set of all negative rational numbers. It is clear that 0* is a cut (see Step 1). Verify that the axioms for addition hold in R, with 0* playing the role of 0. 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 leastupper-bound property. Step 2 Establish R as an ordered set. Hint Define α < β. Step 5 Having proved that the addition defined in Step 4 satisfies Axioms (A) of Definition 1.3.1, it follows that Proposition 1.3.3 is valid in R. Prove one of the requirements of Definition 1.3.6: If α, β, γ in R and β < γ then, α + β < α + γ and that α > 0* if and only if −α < 0*. 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 Step 3 Prove that the ordered set R has the least-upper-bound property. 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. Step 9 1.4.2 Theorem: Archimedean Property of R and Q is Dense in R Prove (a) the archimedean property of R. If x, y ∈ R, and x > 0, then there is a positive integer n such that nx > y. (b) that Q is dense in R, i.e., between any two real numbers, there is a rational one. In other words, if x, y ∈ R, and x < y, then ∃ p ∈ Q such that x < p < y.
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