I have a rather simple question here which however I feel needs a lot of background info to ask properly. So I apologize in advance!
I'm self-studying with Fitzpatrick's Advanced Calculus and am working on a problem from the very first section, where no calculus has been introduced yet:
Define .
a. Show that is an upper bound for and therefore, by the completeness axiom, has a least upper bound that we denote by .
b. Show that if , then we can choose a suitably small positive number such that is also an upper bound for , thus contradicting the choice of as the least upper bound of .
c. Show that if , then we can choose a suitably small positive number such that belongs to , thus contradicting the choice of as an upper bound of .
d. Use parts (b) and (c) and the positivity axioms for to conclude that .
The problem was asked previously in this thread: Upper Bound.
From the hints on that thread I think I have answered parts a-c, however, I have a question on part d. Parts b and c showed that and , respectively. So it seems obvious that must equal , but I am confused by the reference to the positivity axioms. As stated in the books, they are:
1. If and are positive, then and are also positive.
2. For a real number , exactly one of the following three alternatives is true: is positive, is positive, .
How do these allow us to conclude that $b^2=c$? Sorry for the involved question, it's probably really simple but I have no one to ask.
Since people here are so helpful, I thought maybe I'd ask to make sure I have parts b and c right. This is my proof for part b; c is similar. I guess where I get confused is what allows us to assume that we can always choose an such that , since the concept of limit has not been developed yet.
b. Show that if , then we can choose a suitably small positive number such that is also an upper bound for , thus contradicting the choice of as the least upper bound of .
Suppose . Then can be written as for some positive number . Then So will be larger than when , i.e. when . So if we choose such that , then we get . Then by the definition of , for all . So is an upper bound for which is a smaller upper bound than . Since supposing that contradicts our assumption that , we must have .