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

Theorem: If $G$ is an abelian ordered 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 non-zero elements $x,y\in E$ are comparable if there are $m,n\in\mathbb{N}$ such that $|x|<m|y|$ and $|y|<n|x|$, where $|a|=\max\{a,-a\}$ for every $a\in E$.
• Let $E/ K$ be an extension of ordered fields, where the order on $E$ restricted to $K$ coincides with the order 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 there is no proper archimedean extension of $K$.

Someone knows any good references for the proof of this result? The more the better.

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.