The author analyzes the convergence of the standard adaptive loop, i.e., solve, estimate, mark, and refine in solving by a finite element method linear elliptic boundary value problems. He uses the essential fact that the Galerkin solutions to such problems are Cauchy sequences in appropriate functional spaces. In order to exemplify, he considers in turn important examples, namely: a second order elliptic and self adjoint Dirichlet problem, a 2D classical biharmonic problem, a 3D eddy current problem and 2D and 3D Stokes problems.