I need a reference (different from Hahn's 1907 paper) for the following result.



> 
> **Theorem:** If $G$ is a totally ordered abelian group, then the field $\mathbb{R}((G))$ is archimedean complete.
> 
> 
> 


* $\mathbb{R}((G))$ consists of all the functions $f:G\to\mathbb{R}$ such that 
$\{g\in G:f(g)\neq0\}$ is well-ordered.
* Let $E$ be an ordered field. Two nonzero elements $x,y\in E$ are *comparable* if there exist $m,n\in\mathbb{N}$ such that $|x|<m|y|$ and $|y|<n|x|$ (where $|a|$ is defined as $\max\{a,-a\}$).
* Let $E/ K$ be an extension of ordered fields, where the order on $E$ restricted to $K$ coincides with that of $K$. We say that $E$ is an *Archimedean extension of $K$* if for every $x\in E$, there exists $y\in K$ such that $x$ and $y$ are comparable in $E$.
* A field $K$ is *Archimedean complete* if it has no proper archimedean extension fields.


Does anyone know a good reference for the proof of this result? The more the better.



Archimedean completeness of some fields
There are many such proofs in the literature. The completeness for Hahn fields follows from the completeness for Hahn groups, since every Hahn field is a Hahn group. For a simple proof of the latter, see pp. 862-863 of *Note on Hahn's Theorem on Ordered Abelian Groups*, A.H. Clifford, Proc. Am Math. Soc. 5 (1954), pp. 860-863. Many other references can be found in the following question of mine on Mathoverflo [Hahn's Embedding Theorem and the oldest open question in set theory](https://mathoverflow.net/questions/128935/hahns-embedding-theorem-and-the-oldest-open-question-in-set-theory).


