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 http://mathoverflow.net/questions/128935/hahns-embedding-theorem-and-the-oldest-open-question-in-set-theory.