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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
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
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