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 alsocontractive sequence (http://planetmath.org/ContractiveSequence).

Theorem.  Every sequence of bounded variation isconvergent (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 ofseries, that the sequence (1) is a Cauchy sequence and thusconverges. □

One kind of sequences of bounded variation is formed by thebounded monotonic sequences of real numbers (those sequencesare convergent, as is well known).  Indeed, if (1) is a boundedand 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) ofthe series ${\sum }_{i=1}^{\mathrm{\infty }}|a_{i+1}-a_i|$ with nonnegativeterms are bounded.  Therefore the last series is convergent, i.e.our sequence (1) is of bounded variarion.