# Sequence, Supremum, Infimum

• March 11th 2010, 07:31 PM
Slazenger3
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.
• March 11th 2010, 08:06 PM
davismj
Quote:

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?
• March 11th 2010, 08:22 PM
Tinyboss
If sup A and inf B are not equal, then they differ by some positive distance, right?
• March 12th 2010, 04:26 AM
HallsofIvy
Quote:

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 $sup A\le inf B$