contractive sequence
The sequence
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 inequality
d(an,an+1)≦r⋅d(an-1,an) | (2) |
is true.
We will prove the
Theorem. If the sequence (1) is contractive, it is
a Cauchy sequence.
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 inequality we get
d(am,an) | ≦d(am,am+1)+d(am+1,am+δ) | ||
≦d(am,am+1)+d(am+1,am+2)+d(am+2,am+δ) | |||
… | |||
≦d(am,am+1)+d(am+1,am+2)+d(am+2,am+3)+…+d(an-1,an). |
Now the contractiveness gives the inequalities
d(a1,a2)≦rd(a0,a1), |
d(a2,a3)≦rd(a1,a2)≦r2d(a0,a1), |
d(a3,a4)≦rd(a2,a3)≦r3d(a0,a1), |
… |
d(am,am+1)≦rmd(a0,a1), |
… |
d(an-1,an)≦rn-1d(a0,a1), |
by which we obtain the estimation
d(am,an) | ≦d(a0,a1)(rm+rm+1+…+rm+δ-1) | ||
=d(a0,a1)rm(1+r+r2+…+rδ-1) | |||
=d(a0,a1)rm1-rδ1-r | |||
<d(a0,a1)rm1-r. |
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 ℝ, the
sequence 1,12,13,… converges 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 variation (http://planetmath.org/SequenceOfBoundedVariation).
References
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 |