1. ## Sequence, Supremum, Infimum

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.

2. Originally Posted by Slazenger3
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.
What do you have so far?

3. If sup A and inf B are not equal, then they differ by some positive distance, right?

4. Originally Posted by Tinyboss
If sup A and inf B are not equal, then they differ by some positive distance, right?
Yes, that's true of any two numbers! I think you also need to show that $\displaystyle sup A\le inf B$