  The dimension datum of a closed subgroup of a compact Lie group is a sequence
by assigning the invariant dimension of each irreducible representation
restricting to the subgroup. We prove that any sequence of dimension data
contains a converging sequence with limit the dimension datum of a subgroup
interrelated to subgroups giving this sequence. This rigidity has an immediate
corollary that the space of dimension data of closed subgroups in a given
compact Lie group is sequentially compact.
