Priciple of Mathematical Analysis
1.10 Definition: Least-upper-bound property
If $E \subset S$ and $E$ is bounded above, then $\sup{E}$ exists in $S$.
1.11 Theorem
S is a ordered list set with least-upper-bound property. Then, $\forall E \subset S$, if $E$ is nonempty and bounded above, then $\sup{E}$ exists in S.
Proof
1.19 Theorem
There exists an ordered field $\mathbb{R}$ which has the least-upper-bound property.
Proof:
1.20 Theorem
(a)[archimedean property] If $x \in \mathbb{R}, y \in \mathbb{R}, \text{ and } x > 0$, then there exists a positive integer $n$ such that $nx > y$.
(b) If $x \in \mathbb{R}, y \in \mathbb{R}, \text{ and } x < y$, then there exists a rational number $p$ such that $x < p < y$.
Proof: