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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 theoryHahn's Embedding Theorem and the oldest open question in set theory.

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.

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.

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Philip Ehrlich
Philip Ehrlich
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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.