contractive sequence

 $\displaystyle a_{0},a_{1},a_{2},\ldots$ (1)

in a metric space $(X,d)$ is called contractive, iff there is a real number $r\in(0,1)$ such that for any positive integer $n$ the inequality  $\displaystyle d(a_{n},a_{n+1})\leqq r\!\cdot\!d(a_{n-1},a_{n})$ (2)

is true.

We will prove the

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

 $\displaystyle d(a_{m},a_{n})$ $\displaystyle\leqq d(a_{m},a_{m+1})+d(a_{m+1},a_{m+\delta})$ $\displaystyle\leqq d(a_{m},a_{m+1})+d(a_{m+1},a_{m+2})+d(a_{m+2},a_{m+\delta})$ $\displaystyle\ldots$ $\displaystyle\leqq d(a_{m},a_{m+1})+d(a_{m+1},a_{m+2})+d(a_{m+2},a_{m+3})+% \ldots+d(a_{n-1},a_{n}).\par$

Now the contractiveness gives the inequalities

 $d(a_{1},a_{2})\leqq rd(a_{0},a_{1}),$
 $d(a_{2},a_{3})\leqq rd(a_{1},a_{2})\leqq r^{2}d(a_{0},a_{1}),$
 $d(a_{3},a_{4})\leqq rd(a_{2},a_{3})\leqq r^{3}d(a_{0},a_{1}),$
 $\ldots$
 $d(a_{m},a_{m+1})\leqq r^{m}d(a_{0},a_{1}),$
 $\ldots$
 ${d(a_{n-1},a_{n}})\leqq r^{n-1}d(a_{0},a_{1}),$

by which we obtain the estimation

 $\displaystyle d(a_{m},a_{n})$ $\displaystyle\leqq d(a_{0},a_{1})(r^{m}+r^{m+1}+\ldots+r^{m+\delta-1})$ $\displaystyle=d(a_{0},a_{1})r^{m}(1+r+r^{2}+\ldots+r^{\delta-1})$ $\displaystyle=d(a_{0},a_{1})r^{m}\frac{1-r^{\delta}}{1-r}$ $\displaystyle

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

 $d(a_{m},a_{n})<\varepsilon\mbox{ for each }m>M$

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 $\mathbb{R}$, the sequence  $1,\frac{1}{2},\frac{1}{3},\ldots$converges  to 0 and hence is Cauchy, but for it the ratio

 ${|a_{n}-a_{n+1}|:|a_{n-1}-a_{n}}|\;=\;1-\frac{2}{n+1}$

tends to 1 as  $n\to\infty$.

References

• 1 Paul Loya: .  A Course in Undergraduate Analysis, Fall 2006.  Available in http://www.math.binghamton.edu/dennis/478.f07/EleAna.pdf
Title contractive sequence ContractiveSequence 2014-11-30 16:45:13 2014-11-30 16:45:13 pahio (2872) pahio (2872) 13 pahio (2872) Theorem