# Thread: Least upper bound and positivity axioms

1. ## Least upper bound and positivity axioms

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 $\displaystyle S\equiv \{x\mid x\in\mathbb{R},\,x\geq0,\,x^2<c\}$.
a. Show that $\displaystyle c+1$ is an upper bound for $\displaystyle S$ and therefore, by the completeness axiom, $\displaystyle S$ has a least upper bound that we denote by $\displaystyle b$.
b. Show that if $\displaystyle b^2>c$, then we can choose a suitably small positive number $\displaystyle r$ such that $\displaystyle b-r$ is also an upper bound for $\displaystyle S$, thus contradicting the choice of $\displaystyle b$ as the least upper bound of $\displaystyle S$.
c. Show that if $\displaystyle b^2<c$, then we can choose a suitably small positive number $\displaystyle r$ such that $\displaystyle b+r$ belongs to $\displaystyle S$, thus contradicting the choice of $\displaystyle b$ as an upper bound of $\displaystyle S$.
d. Use parts (b) and (c) and the positivity axioms for $\displaystyle \mathbb{R}$ to conclude that $\displaystyle b^2=c$.