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.

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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.

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What do you have so far?

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

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Originally Posted by Tinyboss

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

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Yes, that's true of any two numbers! I think you also need to show that