In each of the following exercises, verify the given statement carefully, proceeding step...

In each of the following exercises, verify the given statement carefully, proceeding step by step. Validate each step that involves an inequality by using some statement found in this section.

a) Let R+ represent the collection of positive real numbers. Prove that R+ satisfies the following two properties.

i) For each x ∈ R, one and only one of the following holds:

ii) Given x, y ∈ R+, both x + y and x + y belong to R+.

b) Suppose that R contains a subset R+ (not necessarily the set of positive numbers) which satisfies properties i) and ii). Define x ≺ y by y − x ∈ R+. Prove that Postulate 2 holds with ≺ in place of <.