---
tag: 0AXWET80YW
kind: statement
role: axiom
name: Completeness axiom
aliases: [least-upper-bound-property, supremum-property, dedekind-completeness]
msc: [26A03]
uses: [08J2M1JYAG, 0B5H2V4JW8, 0BTMQ04556]
sources:
  - url: https://en.wikipedia.org/wiki/Least-upper-bound_property
    locator: Statement for real numbers
same_as:
  wikidata: Q1254734
  wikipedia: Least-upper-bound property
---

## Statement

Let $S$ be a nonempty subset of the [[real numbers|08J2M1JYAG]] that is [[bounded above|0B5H2V4JW8]].
Then $S$ has a [[supremum|0BTMQ04556]] in $\mathbb{R}$.

## Remarks

In the axiomatic approach, this is the axiom that distinguishes $\mathbb{R}$ from the other ordered fields. In a construction of $\mathbb{R}$, it is a theorem.
The ordered field $\mathbb{Q}$ does not have this property: $\{q \in \mathbb{Q} : q^2 < 2\}$ has no supremum in $\mathbb{Q}$.
The axiom gives the [[existence of the infimum|0EDH403M21]] and the [[Archimedean property of the reals|0A3GWZ86AY]].

## Explanations

### L1 human

The real line has no holes.
If a set of real numbers has a ceiling, then there is a lowest ceiling, and this lowest ceiling is again a real number.