- #1

SpringPhysics

- 107

- 0

## Homework Statement

If a sequence {x

_{n}} in ℝ

^{n}satisfies that sum || x

_{n}- x

_{n+1}|| for n ≥ 1 is less than infinity, then show that the sequence is Cauchy.

## Homework Equations

The triangle inequality?

## The Attempt at a Solution

|| x

_{m}- x

_{n}|| ≤ || Ʃ (x

_{i+1}- x

_{i}) from i=n to m-1||

Using the triangle inequality and the given condition, I only get that the norm is less than infinity. I do not know how to transform this into an ε argument. Is there a property of finite sums of telescoping norms that would help?