- #1
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I am reading Ethan D. Bloch's book: The Real Numbers and Real Analysis ...
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
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