Completeness Equivalences

Completeness equivalences. For the ordered field $\mathbb R$, the following statements are equivalent:

1. Least-upper-bound property: every nonempty bounded-above subset of $\mathbb R$ has a supremum.
2. Cauchy completeness: every Cauchy sequence in $\mathbb R$ converges in $\mathbb R$.
3. Monotone convergence: every bounded monotone sequence in $\mathbb R$ converges.
4. Nested-interval property: if $(I_n)$ is a nested sequence of nonempty closed bounded intervals whose lengths tend to $0$, then $\bigcap_{n=1}^{\infty}I_n$ contains exactly one point.