sequence of bounded variation

The sequence

a1,a2,a3, (1)

of complex numbersMathworldPlanetmathPlanetmath is said to be of bounded variationMathworldPlanetmath, iff it satisfies


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


from which we see that


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

One kind of sequences of bounded variation is formed by the boundedPlanetmathPlanetmathPlanetmathPlanetmath 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

aiai+1 for each i,


i=1n|ai+1-ai|=i=1n(ai+1-ai)=an+1-a1. (2)

The boundedness of (1) thus implies that the partial sums (2) of the series i=1|ai+1-ai| with nonnegative terms are bounded.  Therefore the last series is convergent, i.e. our sequence (1) is of bounded variarion.


  • 1 Paul Loya: Amazing and Aesthetic Aspects of Analysis: On the incredible infinite.  A Course in Undergraduate Analysis, Fall 2006.  Available in
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