Completeness equivalences

The real numbers $\mathbb{R}$ are complete . In practice, completeness can be expressed in several equivalent ways.

Completeness equivalences (standard list): The following statements are equivalent (each can be taken as a definition of completeness of $\mathbb{R}$):

• Least upper bound property: Every nonempty set $E\subseteq\mathbb{R}$ that is bounded above has a supremum in $\mathbb{R}$.
• Cauchy completeness: Every Cauchy sequence in $\mathbb{R}$ converges to a real number.
• Nested interval property: If $I_n=[a_n,b_n]$ are nested closed intervals with $I_{n+1}\subseteq I_n$ and $b_n-a_n\to 0$, then $\bigcap_{n=1}^\infty I_n$ consists of exactly one point.
• Monotone convergence: Every bounded monotone sequence in $\mathbb{R}$ converges.

These equivalences explain why different-looking arguments (suprema, Cauchy sequences, nested intervals, monotone sequences) are interchangeable in real analysis.

Proof sketch (outline of implications):

• LUB $\Rightarrow$ monotone convergence: apply $\sup$ to $\{x_n\}$ for an increasing bounded sequence.
• Monotone convergence $\Rightarrow$ Cauchy completeness: given a Cauchy sequence $(x_n)$, define $a_n=\inf_{k\ge n} x_k$ and $b_n=\sup_{k\ge n} x_k$; then $(a_n)$ increases, $(b_n)$ decreases, and Cauchy-ness forces $b_n-a_n\to 0$, yielding a common limit trapped between them.
• Cauchy completeness $\Rightarrow$ nested interval property: choose $x_n\in I_n$; shrinking diameters force $(x_n)$ to be Cauchy, hence convergent; closedness gives the limit lies in every $I_n$.
• Nested interval property $\Rightarrow$ LUB: for $E$ bounded above, build nested intervals $[a_n,b_n]$ with $a_n\in E$ and $b_n$ upper bounds, shrinking so that the unique intersection point is the least upper bound.