Linked from

The 11 pages that link to Completeness of the real numbers, each with the reason it gives.

• Cauchy sequenceRelated: It underwrites the fact that real Cauchy sequences cannot escape the real numbers.

• Real analysisRelated: Completeness guarantees limits that rational numbers alone cannot supply.

• Intermediate value theoremRelated: A least-upper-bound argument supplies a boundary point where continuity forces the intermediate value.

• Dedekind cutRelated: The cut construction supplies the gaps whose absence gives the reals completeness.

• Nonstandard analysisRelated: Completeness distinguishes the standard real field from its nonstandard extensions.

• Squeeze theoremRelated: Completeness underlies the real-number framework in which standard limit theorems hold.

• Upper boundRelated: This completeness property guarantees that bounded real sets have suprema.

• Monotone convergence theoremRelated: The theorem is a standard consequence of this foundational axiom.

• Cantor–Dedekind axiomRelated: Completeness distinguishes the real line from the rational line.

• Connected relationRelated: It is the preference-theory counterpart of connectedness, often including self-comparison.

• Standard part functionRelated: Completeness ensures each finite hyperreal lies infinitesimally close to a real number.