Completeness of the Real Numbers

Last updated Nov 1, 2022

We can state this in the following 6 ways all of which are equivalent:

1. $\mathbb{R}$ with its usual Total Ordering $\leq$ is a Complete Ordering
2. Every Cauchy Sequence of $\mathbb{R}$ converges in $\mathbb{R}$.
3. $\mathbb{R}$ satisfies the Nested Intervals Theorem
4. $\mathbb{R}$ satisfies the Bounded Monotone Convergence Theorem.
5. $\mathbb{R}$ satisfies the Bolzano-Weierstrass Theorem.
6. $\mathbb{R}$ satisfies the Intermediate Value Theorem

# Approach 1: Completion by Cauchy Sequences

TODO In this approach we define a notion of Cauchy Sequences on $\mathbb{Q}$ and represent $\mathbb{R}$ as Cauchy Sequences on $\mathbb{Q}$.