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## Sequence of a bounded variation

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Tolaso J Kos
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### Sequence of a bounded variation

Let $x_n$ be a sequence in the metric space $(X, d)$ . We define $x_n$ to be of a bounded variation if:

$$\sum_{n=1}^{\infty}d\left ( x_n, x_{n+1} \right )<+\infty$$

Prove the following:
a) If $x_n$ is of a bounded variation then it is a standard / basic sequence. (therefore bounded). Does the converse hold?

b) If $x_n$ is a standard/ basic sequence , then there exists a subsequence of a bounded variation.

c) If every subsequence of $x_n$ is of a bounded variation , then $x_n$ is a basic/ standard sequence.
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