What does "convergence" mean for Thinning? I've been using Thinning recently on line-like structures in an attempt to generate skeleton-like chains of pixels connected through their Moore neighborhoods.  

The description of Thinning says that the algorithm "finds the skeletons of foreground regions in image by applying morphological thinning until convergence".  What does "convergence" mean?  If I start with a cluster of pixels connected through their Moore neighborhoods, what am I guaranteed after "convergence"?  Will the product always be a chain of pixels also connected through their Moore neighborhoods?
The thinning operation become idempotent after a number of iterations, which means the result does not change anymore - in other word, the algorithm have converged to a solution.