Prove the following claim, using proof by induction

Given the existence of a set, R, which is the set of real numbers, with the operations of addition and multiplication, and the order relation < on R such that, for all x, y, z ∈ R, there are the following axioms:

Axiom 1: (x + y) + z = x + (y + z); (x * y) * z = x * (y * z)

Axiom 2: x + y = y + x; x * y = y * x

Axiom 3: x * (y + z) = (x * y) + (x * z)

Axiom 4: There is a unique element 0 ∈ R such that 0 + x = x  for all x ∈ R.

Axiom 5: For each x ∈ R, there is a unique y ∈ R such that x + y = 0, and we write y = -x.

Axiom 6: There is a unique element 1 ∈ R such that x * 1 = x for all x ∈ R and 0 ≠ 1.

Axiom 7: For each x ∈ R, with x ≠ 0, there is a unique element y ∈ R such that x * y = 1, and we write y = 1/x.

Axiom 8: x < y implies x + z < y + z

Axiom 9: x < y and y < z implies x < z

Axiom 10: For x, y ∈ R, exactly one of the following is true: x < y, y < x, or x = y.

Axiom 11: x < y and z > 0 implies xz < yz

 

Part , –>  Let a and c be real numbers, with a < c. Using the axioms of the real number system given in part A, prove there exists a real number, b, so that a < b < c.

 

Part 2 –> Create two subsets of the real numbers, C and D, where C is unbounded and D is bounded and infinite.

1. State the supremum and infimum of both C and D.

2. State the supremum and infimum of C U D, or explain why they do not exist.

3. State the supremum and infimum of C ∩ D, or explain why they do not exist.

 

Part 3-> Prove the following claim, using proof by induction. Show your work.

Let d be the day you were born plus 7 (e.g., if you were born on March 24, d = 24 + 7). If a = 2d + 1 and b = d + 1, then an – b is divisible by d for all natural numbers n.

 

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