In: Advanced Math
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.)