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.
Philip Ehrlich
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