Let R be a ring and I an ideal. For an R-module M we define the completion M^* of M to be the limit of the modules M/I^nM. We say M is complete if the natural map M —> M^* is an isomorphism.
Then you ask yourself: Is the completion M^* complete? The answer is no in general, and I just added an example to the chapter on examples in the stacks project.
But… it turns out that if I is a finitely generated ideal in R then M^* is always complete. See the section on completion in the algebra chapter. I’ve found this also on the web in some places… and apparently it occurs first (?) in a paper by Matlis (1978). Any earlier references anybody?