READING:     II     2     O4                    Back to reading:     II     2     O3
begin quote ita"
CHAPTER II [:] The Real Number System [...] 2. ORDER. "end quote
Please get the book and read along.    
topic index     contents

O4:       THE PRODUCT OF THE GREATER POSITIVES IS GREATER.

For any real numbers   a, b, c, d such that    ab0,   and   cd0   we have   acbd
   
  By the meaning  
  of  
  the symbol ,  
  we need to look at the quantity   ac-bd   and show that it is in .  
   
   
   
   
  Furthermore, the given inequalities lead us to look at   a-b   and   c-d.  
   
   
   
   
  Experience suggests that we look at the sum of   (a-b)c   and   b(c-d).  
   
   
   
   
   
  (1)  
  (a-b)c  
  =  
  (a+(-b))c  
  by the definition of subtraction  
   
   
  =  
  ac+(-b)c  
  by the distributivity of muliplication over addition  
   
   
  =  
  ac+(-1·b)c  
  by F10  
   
   
  =  
  ac+(-1)(bc)  
  by the associativity of multiplication  
   
   
  =  
  ac+(-(bc))  
  by F10  
   
   
  =  
  ac-bc  
  by the definition of subtraction  
   
   
   
   
   
  (2)  
  b(c-d)  
  =  
  bc-bd  
  by a sequence of steps very much like those above  
   
   
   
   
   
  (3)  
  (a-b)c + b(c-d)  
  =  
  (ac-bc) + (bc-bd)  
  by the two results above  
   
   
  =  
  ac-bd  
  by a sequence of steps using the same properties we have been using  

By the given facts, the first term on the left hand side of the equals sign in (3),   (a-b)c,   is in .
Furthermore, the second term,   b(c-d),   is either zero or in .
Therefore,   ac-bd   and we have the desired result.

on to O5