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.

Welcome to mathimatikoi.org forum; Enjoy your visit here.

## Sequence of a bounded variation

- Tolaso J Kos
- Administration team
**Articles:**2**Posts:**860**Joined:**Sat Nov 07, 2015 6:12 pm**Location:**Larisa-
**Contact:**

### Sequence of a bounded variation

**Imagination is much more important than knowledge.**