Math 2.0

Real functions The real line Theorem

Completeness of the real numbers

Builds on Archimedean ordered fieldCompleteness axiomCauchy sequenceMonotone sequenceMonotone convergence theorem for sequencesBolzano–Weierstrass theorem

Statement

Let FF be an Archimedean ordered field. Then the following five conditions on FF are equivalent.

  1. Each nonempty subset of FF that is bounded above has a supremum in FF (the least-upper-bound property).
  2. Each Cauchy sequence in FF converges in FF.
  3. For each sequence of closed bounded intervals [a1,b1]⊇[a2,b2]⊇⋯[a_1, b_1] \supseteq [a_2, b_2] \supseteq \cdots in FF, some point of FF lies in all of them (the nested intervals property).
  4. Each bounded monotone sequence in FF converges in FF.
  5. Each bounded sequence in FF has a subsequence that converges in FF.

The field ℝ\R of real numbers satisfies all five conditions.

Remarks

The standard proof goes in a cycle. From (1), a bounded increasing sequence converges to the supremum of its terms, which gives (4). From (4), each sequence has a monotone subsequence, which gives (5). A Cauchy sequence is bounded, and if a subsequence converges then the whole sequence converges, so (5) gives (2). From (2), repeated bisection between a point of a set and an upper bound gives a Cauchy sequence. Its limit is the supremum, and the Archimedean property makes the interval lengths tend to 00. For (3), the left endpoints of nested closed intervals have a supremum under (1), and this supremum lies in every interval. Conversely, (3) gives (1) by the same bisection, with the Archimedean property again making the lengths tend to 00. The field ℚ\Q satisfies none of the conditions: the decimal approximations 1,1.4,1.41,1.414,…1, 1.4, 1.41, 1.414, \ldots of 2\sqrt 2 form a Cauchy sequence with no limit in ℚ\Q. The Archimedean hypothesis matters for (2): there are non-Archimedean ordered fields in which each Cauchy sequence converges, but which lack the least-upper-bound property. Conditions (1), (4) and (5) each imply the Archimedean property by themselves. Up to isomorphism, ℝ\R is the only complete ordered field.

Going deeper

For a student

In a course, one of the conditions is taken as an axiom or proved from a construction of ℝ\R, and the others follow as theorems. Dedekind cuts give the least-upper-bound property directly, while Cauchy sequences give condition (2) directly. The Bolzano–Weierstrass theorem is condition (5), and the monotone convergence theorem is condition (4). The intermediate value theorem and the extreme value theorem both rest on completeness, and both fail over ℚ\Q.

Reading path

To understand this page from the ground up, read these 18 pages in order. The foundations come first.

Show the path as a graph
Ordered field, definition, no evidence levelOrdered fielddefinitionReal numbers, object, no evidence levelReal numbersobjectSequence, definition, no evidence levelSequencedefinitionSet bounded above, definition, no evidence levelSet bounded abovedefinitionArchimedean ordered field, definition, no evidence levelArchimedean ordered f…definitionReal sequence, definition, no evidence levelReal sequencedefinitionSubsequence, definition, no evidence levelSubsequencedefinitionSupremum, definition, no evidence levelSupremumdefinitionBounded sequence, definition, no evidence levelBounded sequencedefinitionCauchy sequence, definition, no evidence levelCauchy sequencedefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionMonotone sequence, definition, no evidence levelMonotone sequencedefinitionCompleteness axiom, statement, no evidence levelCompleteness axiomstatementMonotone convergence theorem for sequences, statement, no evidence levelMonotone convergence …statementConvergent sequence, definition, no evidence levelConvergent sequencedefinitionEvery sequence has a monotone subsequence, statement, no evidence levelEvery sequence has a …statementBolzano–Weierstrass theorem, statement, no evidence levelBolzano–Weierstrass t…statementCompleteness of the real numbers, statement, no evidence levelCompleteness of the r…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Ordered field definition
  2. Real numbers object
  3. Sequence definition
  4. Set bounded above definition
  5. Archimedean ordered field definition
  6. Real sequence definition
  7. Subsequence definition
  8. Supremum definition
  9. Bounded sequence definition
  10. Cauchy sequence definition
  11. Limit of a sequence definition
  12. Monotone sequence definition
  13. Completeness axiom statement
  14. Monotone convergence theorem for sequences statement
  15. Convergent sequence definition
  16. Every sequence has a monotone subsequence statement
  17. Bolzano–Weierstrass theorem statement
  18. Completeness of the real numbers statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Completeness_of_the_real_numbers
About this page
TagE3CPZNJZNG
URIm2://E3CPZNJZNG
Kindstatement, theorem
WritingW0 stub
MSC26A03, 12J15
StatusThis copy of the site has no build data. Run m2 status E3CPZNJZNG.
AboutReal numbers, Complete ordered field
UsesArchimedean ordered field, Completeness axiom, Cauchy sequence, Monotone sequence, Monotone convergence theorem for sequences, Bolzano–Weierstrass theorem
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Changed2 November 2026

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