If $\sup A < \sup B$, does $B$ contain an upper bound of $A$?

Question

If A and B are nonempty subsets of $\mathbb{R}$ and $\sup A < \sup B$, does $B$ contain an upper bound of $A$?

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So, the trivial case is if supB is contained in B. The Archimedean property would be my first guess for the supB not contained in B part. But how would you construct an argument for that?

Answer 1

Hint Let $\epsilon = \frac{\sup(B)-\sup(A)}{2}$.

By the definition of $\sup(B)$ there exists an element $b \in B$ such that $b \geq \sup(B) -\epsilon$....