Every interval (a,b) contains both rational and irrational numbers

[Math Amateur]

I am reading Chapter 1:"Real Numbers" of Charles Chapman Pugh's book "Real Mathematical Analysis.

I need help with the proof of Theorem 7 on pages 19-20.

Theorem 7 (Chapter 1) reads as follows:
View attachment 3828
View attachment 3829In the above proof, Pugh writes:

" ... ... The fact that \(\displaystyle a \lt b\) implies the set B \ A contains two distinct rational numbers, say \(\displaystyle r, s\). ... ... "

Can someone help me to understand exactly how it follows that \(\displaystyle a \lt b\) implies the set B \ A contains two distinct rational numbers, say \(\displaystyle r, s\)?

Peter***NOTE***

Since Theorem 7, Chapter 1, mentions cuts, i am providing Pugh's definition of a Dedekind cut, as follows:View attachment 3830

 

[Euge]

Peter, could you please explain the meaning of the notation $A|A'$?

 

[Math Amateur]

Peter, could you please explain the meaning of the notation $A|A'$?

Sorry Euge, I should have included that notation after the definition of a cut in \(\displaystyle \mathbb{Q}\) ... ... my apologies ...

A cut in \(\displaystyle \mathbb{Q}\) is a pair of subsets \(\displaystyle A, B\) with the three conditions shown above in my post ... the Dedekind cut is denoted \(\displaystyle A|B\) ...

So \(\displaystyle A|A\)' is a Dedekind cut involving the two sets \(\displaystyle A\) and \(\displaystyle A'\)

Hope that helps ...

Peter