Suppose that A and B are nonempty sets of real numbers and that for every number x in A and every number y in B we have x < y. Prove that the following conditions are equivalent:

(1) We have sup A = inf B

(2) There exists a sequence (X sub n) in the set A and a sequence (Y sub n) in the set B such that (Y sub n)-(X sub n) approaches 0 as n approaches infinity.