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 $?
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Did you mean that $A$ has an upper bound and that $B$ has a lower bound? Also, you should make your own attempt at the problem very clear. Otherwise, the question will get closed. – Abhijeet Vats Nov 19 '20 at 20:16
