  We show that any set of quotients with fixed Chern classes of a given
coherent sheaf on a compact Kaehler manifold is bounded in a sense which we
define. The result is proved by adapting Grothendieck's boundedness criterium
expressed via the Hilbert polynomial to the Kaehler set-up. As a consequence we
obtain the compactness of the connected components of the Douady space of a
compact Kaehler manifold.
