Reference no: EM13136688

1. Using the axioms of real numbers prove the following theorem: 
8x; y 2 R  (x < y) ) (y < x) _ (y = x) 

 
(Note: This is the theorem that allows us to replace x 6 < y with x  y.) 
2. Let x; y 2 R and  2 R 
+ 
. Prove the following theorem: 
If x  y +  for any  > 0 then x  y. 
For this proof you may assume any theorems stated in lectures. 
Hint: Write the theorem as a conditional logic statement (using any necessary quanti er(s)) 
and then consider an indirect proof. 
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