# sequence of bounded variation

The sequence

 $\displaystyle a_{1},a_{2},a_{3},\ldots$ (1)
 $\sum_{n=1}^{\infty}|a_{n}\!-\!a_{n+1}|\;<\;\infty.$

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 $m, we form the telescoping sum

 $a_{m}-a_{n}=\sum_{i=m}^{n-1}(a_{i}-a_{i+1})$

from which we see that

 $|a_{m}-a_{n}|\;\leqq\;\sum_{i=m}^{n-1}|a_{i}-a_{i+1}|.$

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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