sequence of bounded variation
Cf. function of bounded variation. See also
contractive sequence (http://planetmath.org/ContractiveSequence).
Theorem. Every sequence of bounded variation is
Proof. Let’s have a sequence (1) of bounded variation. When , we form the telescoping sum
from which we see that
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
The boundedness of (1) thus implies that the partial sums (2) of the series with nonnegative terms are bounded. Therefore the last series is convergent, i.e. our sequence (1) is of bounded variarion.
|Title||sequence of bounded variation|
|Date of creation||2014-11-28 21:01:47|
|Last modified on||2014-11-28 21:01:47|
|Last modified by||pahio (2872)|