contractive sequence

The sequenceMathworldPlanetmath

a0,a1,a2, (1)

in a metric space (X,d) is called contractive, iff there is a real number r(0,1) such that for any positive integer n the inequalityMathworldPlanetmath

d(an,an+1)rd(an-1,an) (2)

is true.

We will prove the

Theorem.  If the sequence (1) is contractive, it is a Cauchy sequenceMathworldPlanetmathPlanetmath.

Proof.  Suppose that the sequence (1) is contractive. Let ε be an arbitrary positive number and m,n some positive integers from which e.g. n is greater than m, n=m+δ.

Using repeatedly the triangle inequalityMathworldMathworldPlanetmathPlanetmath we get

d(am,an) d(am,am+1)+d(am+1,am+δ)

Now the contractiveness gives the inequalities


by which we obtain the estimation

d(am,an) d(a0,a1)(rm+rm+1++rm+δ-1)

The last expression tends to zero as m.  Thus there exists a positive number M such that

d(am,an)<ε for each m>M

when n>m.  Consequently, (1) is a Cauchy sequence.

Remark.  The assertion of the Theorem cannot be reversed. E.g. in the usual metric of , the sequence  1,12,13,convergesPlanetmathPlanetmath to 0 and hence is Cauchy, but for it the ratio

|an-an+1|:|an-1-an|= 1-2n+1

tends to 1 as  n.

Cf. sequences of bounded variationMathworldPlanetmath (


  • 1 Paul Loya: Amazing and Aesthetic Aspects of AnalysisMathworldPlanetmath: On the incredible infiniteMathworldPlanetmathPlanetmath.  A Course in Undergraduate Analysis, Fall 2006.  Available in
Title contractive sequence
Canonical name ContractiveSequence
Date of creation 2014-11-30 16:45:13
Last modified on 2014-11-30 16:45:13
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 13
Author pahio (2872)
Entry type Theorem