Exercise 8, § 3

Let A be a Noetherian semi-local ring, n its nilradical (which is the largest nilpotent ideal of A) and m its Jacobson radical. Show that, if A (with the m-adic topology) is such that A/n is complete, then A is complete. (Reduce it to the case where n is generated by a single element c such that $c^2 = 0$; using Exercise 9 of General Topology, Chapter III, § 3, reduce it to showing that n = Ac is complete and use the fact that n is an (A/n)-module.)