Let x,y ∈ R satisfy x < y. Prove that there exists a q ∈ Q...

Let x,y ∈ R satisfy x < 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 the set of all integers, show that A is a non-empty subset of R with an upper bound in R. (Hint: Use the Archimedean Property to show that A ≠ ∅.)
By the Completeness Axiom, A has a least upper bound in R, which we shall denote by m. Show that m ∈ A. (Hint: Refer to Problem 3 of Homework Assignment 3.)
Finally, show that x < m/n < y. (Hint: It is immediate from Step 3 that m/n < y. To show that x < m/n, assume that m/n ≤ x and then derive a contradiction.)

Expert Solution

Colby Messinger answered 2 years ago

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