  In this paper we prove genus bounds for closed embedded minimal surfaces in a
closed 3-dimensional manifold constructed via min-max arguments. A stronger
estimate was announced by Pitts and Rubistein but to our knowledge its proof
has never been published. Our proof follows ideas of Simon and uses an
extension of a famous result of Meeks, Simon and Yau on the convergence of
minimizing sequences of isotopic surfaces. This result is proved in the second
part of the paper.
