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

[Math Amateur]

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

 

[willem2]

According to part(3) of theorem 1.2.9
for all a,b,c ∈ ℕ: if a<b and b<=c then a<c.
b = c implies b<=c , and you can substitute a for c.

 

[Math Amateur]

Thanks for the post, willem2 ...

Appreciate your help ...

Peter