why the completion of a local ring $R$ can be written as an increasing union of $R$-algebras of finite type?
EDIT: Given a ring $R$ that is an algebra over a base ring it is always a filtering union of finite type algebras. Take a system of generators of $R$ over the base ring. The family of finite subsets of this system provides a collection of finite type subalgebras of $R$ whose filtered union is $R$.
(Some considerations on completion via Cauchy sequences deleted.)