  We prove some general results on sequential convergence in Fr\'echet lattices
that yield, as particular instances, the following results regarding a closed
ideal I of a Banach lattice E: (i) If two of the lattices E, I and E/I have the
positive Schur property (the Schur property, respectively) then the third
lattice has the positive Schur property (the Schur property, respectively) as
well; (ii) If I and E/I have the dual positive Schur property, then E also has
this property; (iii) If I has the weak Dunford-Pettis property and E/I has the
positive Schur property, then E has the weak Dunford-Pettis property. Examples
and applications are provided.
