Highest and lowest, exercise.

Question

Let $ A $ and $ B $ be non-empty sets of $ \mathbb R $ where $ A $ is upper bound and $ B $ is lower bound. Suppose that for every $ \epsilon> 0 $, there exist $ x \in A $ and $ y\in B $ such that $ 0 <y-x <\epsilon $. Is it true then that $ \sup A = \inf B $?

Answer 1

For a counterexample, you could take $A = B$