Pick two distinct points p and q in [0,1] and let A = {p} and B = {q}. Cut the interval in half, to get [0,1/2] and [1/2,1] and choose a pair of distinct points from each ({a,b} and {c,d} say). Throw into A one point from each interval (say a and c), and do the same for B (b and d). Repeat...