sequence of bounded variation

The sequence

$a_1,a_2,a_3,\mathrm{\dots }$ (1)

of complex numbers is said to be of bounded variation, iff it satisfies

Cf. function of bounded variation. See also contractive sequence (http://planetmath.org/ContractiveSequence).

Theorem.  Every sequence of bounded variation is convergent (http://planetmath.org/ConvergentSequence).

Proof.  Let’s have a sequence (1) of bounded variation.  When , we form the telescoping sum

$$a_m-a_n=\sum _{i=m}^{n-1}(a_i-a_{i+1})$$

from which we see that

$$|a_m-a_n|\leqq \sum _{i=m}^{n-1}|a_i-a_{i+1}|.$$

This inequality shows, by the Cauchy criterion for convergence of series, that the sequence (1) is a Cauchy sequence and thus converges. □

One kind of sequences of bounded variation is formed by the bounded monotonic sequences of real numbers (those sequences are convergent, as is well known).  Indeed, if (1) is a bounded and e.g. monotonically nondecreasing sequence, then

$$a_i\leqq a_{i+1}\mathit{\hspace{1em}}\text{ for each }i,$$

whence

${\displaystyle \sum _{i=1}^n}|a_{i+1}-a_i|={\displaystyle \sum _{i=1}^n}(a_{i+1}-a_i)=a_{n+1}-a_1.$ (2)

The boundedness of (1) thus implies that the partial sums (2) of the series ${\sum }_{i=1}^{\mathrm{\infty }}|a_{i+1}-a_i|$ with nonnegative terms are bounded.  Therefore the last series is convergent, i.e. our sequence (1) is of bounded variarion.

Title	sequence of bounded variation
Canonical name	SequenceOfBoundedVariation
Date of creation	2014-11-28 21:01:47
Last modified on	2014-11-28 21:01:47
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	7
Author	pahio (2872)
Entry type	Theorem