1. Given the finite set A of open intervals in B3 above, say we have a set G whose elements are the set of all distances from from each x in UA to every other y in UA plus the shortest distance from x to the bounding sphere, whose union contains (n+1)^(n+1) distances. The cardinality is meaningless, but the fact that it is finite is not. Let S be the min of each element in G. And let z be min(S). Begin adding an infinite number of points to A. Then, the first added point will be associated with a corresponding element in S < z by the triangle inequality. An so on, thus we have a construction for our l.p.
The statement that p is a limit point of the number set A means if c > 0 there is a q in A such that q is not equal to p and |p - q| < c.