Show that if a, b∈ R with a < b, then there exists a number x ∈ I with a < x < b

[cooljosh2k2]

Homework Statement

Let I be the set of real numbers that are not rational. (We call
elements of I irrational numbers. Show that if a, b are real
numbers with a < b, then there exists a number x ∈ I with
a < x < b.

The Attempt at a Solution

Im not sure if this is right:

Using the density theorem to the real numbers a/√2 and b/√2, we obtain a rational number such that:

a/√2 < r < b/√2

where r ≠ 0

This gives ---> x = r√2, which is an irrational number because it contains √2, which itself is irrational. This satisfies b < x < a

Is this right? or do i need more in the proof?

 

[ZioX]

Perfect.

For the sake of completeness, you might want to add a proof of r \sqrt{2} not being rational. However, if you have shown this result in class or in your notes you should probably reference it.