1.  Take a finite bounded set and Construct a set of open intervals (maximum epsilon balls) around each point s.t. each ball contains only 1 point. Let G be a set whose elements are the set of all distances from from each x in union A to every other y in union A plus the shortest distance from x to the bounding sphere, for each x, 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.  Begin adding an infinite number of points to A.  Then, the first added point will be associated with a corresponding element in S which will be less then at least one element in G(by 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.