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I have the following criteria for the completeness of a metric space that I want to use in some research paper.

Let $(X,d)$ be a metric space. The following conditions are equivalent:

(1) $X$ is complete.

(2) Any sequence $(x_{n})$ in $X$ such that $\sum\limits_{n\geq0}d(x_{n},x_{n+1})<\infty$ is convergent.

(3) Any sequence $(x_{n})$ in $X$ such that $d(x_{n},x_{n+1})\leq\frac{1}{2^{n}}$ for all $n\in\mathbb{N}$ is convergent.

(4) Any Cauchy sequence in $X$ has a convergent subsequence.

I know it is an elementary result, I know how to prove it, but I want to make a reference to some textbook where this result (or a similar one) can be found. Do you have any suggestion?

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    I haven't heard of this criterion before, but if you know how to prove it (and it seems somewhat straightforward), maybe you can put it in the appendix of your paper? If you want I can try to write up a solution and post it as an answer2016-09-12
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    I know how to prove this, I thought that pointing to some textbook was the easiest and the appropriate thing to do, instead of proving this myself.2016-09-12

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