completeness principle

The completeness principle is a property of the real numbers, and is one of the foundations of real analysis. The most common formulation of this principle is that every non-empty set which is bounded from above has a supremum.

This statement can be reformulated in several ways. Each of the following statements is to the above definition of the completeness principle:

  1. 1.

    The limit of every infiniteMathworldPlanetmath decimal sequenceMathworldPlanetmath is a real number.

  2. 2.
  3. 3.

    A sequence is convergent iff it is a Cauchy SequenceMathworldPlanetmathPlanetmath.

Title completeness principle
Canonical name CompletenessPrinciple
Date of creation 2013-03-22 12:23:06
Last modified on 2013-03-22 12:23:06
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 12
Author mathcam (2727)
Entry type Axiom
Classification msc 54E50
Synonym completeness Axiom
Synonym completeness principle
Synonym least upper bound property
Related topic ConvergentSequence
Related topic ExistenceOfSquareRootsOfNonNegativeRealNumbers
Related topic BoundedComplete