Take a look at Theorem 1.3 of http://people.wku.edu/david.neal/337/Reals.pdf. Let me know if you can't follow it.
Suppose x; y in R are rational numbers with x < y. Prove that there exists an irrational
number z with x < z < y.
Hint: use the Archimedean property. You may use the fact that square root 2 is irrational, and the fact that a rational plus an irrational is an irrational.
I'm not sure what he's looking for. I started by using induction, but I got stuck at the induction step as far as how to prove that. I started with a base step of x=0 and y=1, but when i started looking at the problem more, i'm not sure how to account for fractions or things like that.