  The Douglas-Rachford algorithm is a very popular splitting technique for
finding a zero of the sum of two maximally monotone operators. However, the
behaviour of the algorithm remains mysterious in the general inconsistent case,
i.e., when the sum problem has no zeros. More than a decade ago, however, it
was shown that in the (possibly inconsistent) convex feasibility setting, the
shadow sequence remains bounded and it is weak cluster points solve a best
approximation problem.
  In this paper, we advance the understanding of the inconsistent case
significantly by providing a complete proof of the full weak convergence in the
convex feasibility setting. In fact, a more general sufficient condition for
the weak convergence in the general case is presented. Several examples
illustrate the results.
