  In this paper we shall prove the weak convergence of the associated diffusion
processes of regular subspaces with monotone characteristic sets for a fixed
Dirichlet form. More precisely, given a fixed 1-dimensional diffusion process
and a sequence of its regular subspaces, if the characteristic sets of regular
subspaces are decreasing or increasing, then their associated diffusion
processes are weakly convergent to another diffusion process. This is an
extended result of [13].
