contractive sequence

The 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

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

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 $m,$ $n=m+\delta$ .

Using repeatedly the triangle inequality we get

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

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

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

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 $n\to\infty$ .

Cf. sequences of bounded variation (http://planetmath.org/SequenceOfBoundedVariation).

References

1 Paul Loya: Amazing and Aesthetic Aspects of Analysis: On the incredible infinite. A Course in Undergraduate Analysis, Fall 2006. Available in http://www.math.binghamton.edu/dennis/478.f07/EleAna.pdf

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