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

  1. Jul 16, 2017 #1
    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:



    ?temp_hash=cd0435e2e2ade0b058500f4b7df2618f.png
    ?temp_hash=cd0435e2e2ade0b058500f4b7df2618f.png




    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
     

    Attached Files:

  2. jcsd
  3. Jul 16, 2017 #2
    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.
     
  4. Jul 16, 2017 #3
    Thanks for the post, willem2 ...

    Appreciate your help ...

    Peter
     
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