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
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)
RationalNumber
RealNumber
(none)

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CauchySequence
ComplexNumber
GeneralSequence
AlgebraicNumber
ContinuedFraction
CountableSet
DedekindCut
Denominator
Integer
IrrationalNumber
Numerator
Pythagoras
ReducingFractionsToLowestTerms
RootTwoIsIrrational
TranscendentalNumber
ArgandDiagram
ArithmeticSequence
CategoryMetaTopic
ComplexConjugate
ComplexPlane
DividingComplexNumbers
Euler
GeometricSequence
ImaginaryNumber
MultiplyingComplexNumbers
PolarRepresentationOfAComplexNumber