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.