  In this note we give a quick and simple proof of the existence (and
uniqueness) of Zariski decompositions on surfaces. While Zariski's original
proof employs a rather sophisticated procedure to construct the negative part
of the decomposition, the present approach is based on the idea that the
positive part can be constructed from a maximality condition. It may also be
useful that this approach yields a practical algorithm for the computation of
the positive part.
