completeness principleAxiomOfAnalysis2002-02-19 06:24:542005-06-10 14:59:09Axiom% this is the default PlanetMath preamble. as your knowledge
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% define commands hereThe \emph{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 \PMlinkescapetext{equivalent} to the above definition of the completeness principle:
\begin{enumerate}
\item The limit of every infinite decimal sequence is a real number.
\item Every bounded monotonic sequence is convergent.
\item A sequence is convergent iff it is a Cauchy Sequence.
\end{enumerate}