Completeness Equivalences

Completeness equivalences: For an ordered field satisfying the field axioms and order axioms, the following statements are equivalent (each implies the others):

1. (Least upper bound property) Every nonempty bounded above subset of $\mathbb{R}$ has a supremum (the completeness axiom).
2. (Cauchy completeness) Every Cauchy sequence in $\mathbb{R}$ is a convergent sequence.
3. (Monotone convergence) Every monotone sequence in $\mathbb{R}$ that is bounded converges (see monotone sequence convergence theorem).
4. (Nested intervals) If $(I_n)$ is a nested sequence of nonempty closed intervals with lengths tending to $0$, then $\bigcap_{n=1}^\infty I_n$ consists of exactly one point.