The Main Theorem of a 1943 paper by Zariski states: Let V and V' be varieties, and let f : V' → V be a birational morphism. Assume that V is normal, and let P be a (not necessarily closed) point of V. If f-1(P) contains an isolated point of the same dimension as P, then f-1(P) is a set of finitely many points, all of which have the same residue field. In 1949, Zariski published a short paper, "A simple analytical proof of a fundamental property of birational transformations", Proc. Nat. Acad. Sci. U.S.A. 35, (1949) 62-66, in which he gives a proof of his Main Theorem using completions and valuations of local rings. We will present this proof, assuming only some standard facts from commutative algebra.