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Homesequence of bounded variation

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# sequence of bounded variation

The sequence

$\displaystyle a_{1},a_{2},a_{3},\ldots$ | (1) |

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

$\sum_{{n=1}}^{\infty}|a_{n}\!-\!a_{{n+1}}|\;<\;\infty.$ |

Cf. function of bounded variation. See also
contractive sequence.

Theorem. Every sequence of bounded variation is
convergent.

Proof. Let’s have a sequence (1) of bounded variation. When $m<n$, 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

$a_{i}\leqq a_{{i+1}}\quad\mbox{ for each }i,$ |

whence

$\displaystyle\sum_{{i=1}}^{n}|a_{{i+1}}-a_{i}|=\sum_{{i=1}}^{n}(a_{{i+1}}-a_{i% })=a_{{n+1}}-a_{1}.$ | (2) |

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

# 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

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