  We show that the Gromov-Hausdorff limit of a sequence of leaves in a compact
foliation is a covering space of the limiting leaf which is no larger than this
leaf's holonomy cover. We also show that convergence to such a limit is smooth
instead of merely Gromov-Hausdorff. Corollaries include Reeb's local stability
theorem, part of Epstein's local structure theorem for foliations by compact
leaves, and a continuity theorem of Alvarez and Candel. Several examples are
discussed.
