 Cluster-set matching of bounded sequences

Two bounded real sequences with the same set of subsequential limits admit permutations that match their terms with differences tending to zero. Alternate between taking the least unused index of the first sequence and of the second. Match its value to an unused term of the other sequence near a nearest common cluster point, with additional error at most $1/j$ at stage $j$. Every cluster-point neighbourhood contains infinitely many available terms. The actively chosen indices tend to infinity, whose distance to the common cluster set tends to zero by boundedness. By step $2j$, both lists have included every index up to $j$.