# Reference for the characterization of completeness for metric spaces

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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