I am reading Ethan D. Bloch's book: The Real Numbers and Real Analysis ...(adsbygoogle = window.adsbygoogle || []).push({});

I am currently focused on Chapter 1: Construction of the Real Numbers ...

I need help/clarification with an aspect of Theorem 1.2.9 (6) ...

Theorem 1.2.9 reads as follows:

In the above proof of (6) we read the following:

" ... ... Suppose that ##a \lt b## and ##a = b##. It then follows from Part (3) of this theorem that ##a \lt a## ... ... "

Can someone please explain how Part (3) of Theorem 1.2.9 leads to the statement that ##a \lt b## and ##a = b \Longrightarrow a \lt a## ... ...

... ... ...

Further ... why can't we argue this way ...

... because ##a = b## we can replace ##b## by ##a## in ##a \lt b## giving ##a \lt a## ... which contradicts Part (1) of the theorem ...

Hope someone can help ...

Peter

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# I Properties of 'less than" and "less than or equals"

Have something to add?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**