Problems-
Problem 1- Let b ∈ R. Let D = {x ∈ Q|x < b}. Prove that D has a least upper bound, and that lub D = b. (This fact seems trivial intuitively, but a proof is needed)
Problem 2- Complete the missing parts of Step 2 of the proof of Theorem. That is, let m, n ∈ Z1, and prove that h(m+ n) = h(m) + h(n), that h(mn) = h(m)h(n), and that if m < n then h(m) = h(n).
Problem 3- Let R1 and R2 be ordered fields that satisfy the Laser Upper Bound Property, and let p: R1→ R2 be a function. Suppose that p(x + y) = p(x) + p(y) for all x, y ∈ R1.
(1) Prove that p(0) = 0.
(2) Prove that p(-x) = -p(x) for all x ∈ R1.
Problem 4- Let n ∈ Z. The integer n is even if there is some k ∈ Z such that n = 2k; the integer n is odd if there is some k ∈ Z such that n = 2k + 1. Prove that every integer is either even or odd, but not both.
Problem 5- Prove that-
(1) 1< 2.
(2) 0 < ½ < 1.
(3) If a, b ∈ R and a < b, then a < (a+b)/2 < b.
Theorem - Let R1 and R2 be ordered fields that satisfy the Least Upper Bound Property. Then there is a function f: R1→ R2 that is bijective, and that satisfies the following properties. Let x, y ∈ R1.
a. f(x + y) = f(x) + f(y).
b. f(xy) = f(x)f(y).
c. If x < y, then f(x) < f(y).