With a general sequence we usually have to say what the set is from which the elements of the sequence have been chosen. Usually we are dealing with one of the usual classes of number, namely the rational numbers, the real numbers, or the complex numbers, but the definition of Cauchy Sequence works in any setting where we have a concept of "distance".

A "Cauchy Sequence" is a general sequence with the following property:

You think of a distance, then

I say where to start, then

you find that from then on,

any two elements in the sequence differ by no more than your stated distance.

That sounds convoluted, but the idea is that the sequence can bounce around as much as it likes to start with, but after some point it settles down and all the elements are close. Indeed, you say how close you want them to be, and we find that if we go far enough, everything is within that tolerance.

Equivalence classes of Cauchy sequences of rational numbers are one way to construct the real numbers.

AbsoluteValue Adding2DVectors AddingRealNumbers AddingVectors AlgebraicNumber ArgandDiagram ComplexConjugate ComplexNumber ComplexPlane ContinuedFraction ContinuumHypothesis CountableSet CubeRoot DedekindCut Denominator DifferenceOfTwoSquares DividingComplexNumbers DividingRationalNumbers DividingRealNumbers DividingWholeNumbers Divisor DomainOfAFunction Function FundamentalTheoremOfAlgebra GeorgCantor ImaginaryNumber ImproperFraction Integer IrrationalNumber Logarithm MagnitudeOfAVector MathematicsTaxonomy MixedNumber MultiplyingRationalNumbers MultiplyingRealNumbers NewUserIntroduction NewtonsMethod Numerator PolarRepresentationOfAComplexNumber Quaternion ReducingFractionsToLowestTerms RiemannHypothesis RootTwoIsIrrational RootsOfPolynomials SquareRoot SubtractingRationalNumbers SubtractingRealNumbers TranscendentalNumber TypesOfNumber UncountableSet	(none)			(none)
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Cauchy Sequence

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