Let x,y ∈ R satisfyx < y. Prove that there exists a q ∈Q such that x < q <y.
Strategy for solving the problem
- Show that there exists an n ∈N+ such that 0 < 1/n <y - x.
- Letting A = {k : Z |k < ny}, where Z denotes theset of all integers, show that A is a non-empty subset ofR with an upper bound in R.(Hint: Use the Archimedean Property to show thatA ≠∅.)
- By the Completeness Axiom, A has a least upper boundin R, which we shall denote by m. Showthat m ∈ A. (Hint: Refer toProblem 3 of Homework Assignment 3.)
- Finally, show that x < m/n <y. (Hint: It is immediate from Step 3that m/n < y. To show that x< m/n, assume that m/n ≤x and then derive a contradiction.)
