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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## Sequence of a bounded variation

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### Sequence of a bounded variation

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