Hi guys the analysis continues.
Let
QUESTION 1: Show, without assuming the existence of , that A does not have a maximal element, i.e. prove that if then there exists with .
SOLUTION 1: Let . We need to find delta so small that z > y but . We require
Take , then .
Also it is safe to assume that y < 2. So
So .
We get a small enough delta therefore if we choose i.e. choose . Indeed since .
QUESTION 2: Show too that if has and , then there is with .
SOLUTION 2: This is where I am having trouble. I have let and again assumed that and . Any help would be appreciated
QUESTION 3: A is non-empty and bounded above therefore by the completeness axiom the supremum of A exists. Write x = sup A and show that . [Hint: one way of doing this is to show that both and lead to a contradiction using the previous step.]
SOLUTION 3: I don't really understand how to use the previous steps to solve this part. Again any help would be appreciated.