(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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.

2. Relevant equations

The triangle inequality?

3. 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?

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

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# Show a sequence is Cauchy

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