# 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

$\sum _{i=1}^{n}}|{a}_{i+1}-{a}_{i}|={\displaystyle \sum _{i=1}^{n}}({a}_{i+1}-{a}_{i})={a}_{n+1}-{a}_{1}.$ | (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.

## References

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 |