We will prove the
Proof. Suppose that the sequence (1) is contractive. Let be an arbitrary positive number and some positive integers from which e.g. is greater than .
Using repeatedly the triangle inequality we get
Now the contractiveness gives the inequalities
by which we obtain the estimation
The last expression tends to zero as . Thus there exists a positive number such that
when . Consequently, (1) is a Cauchy sequence.
Remark. The assertion of the Theorem cannot be reversed. E.g. in the usual metric of , the sequence converges to 0 and hence is Cauchy, but for it the ratio
tends to 1 as .
Cf. sequences of bounded variation (http://planetmath.org/SequenceOfBoundedVariation).
|Date of creation||2014-11-30 16:45:13|
|Last modified on||2014-11-30 16:45:13|
|Last modified by||pahio (2872)|