Let $A$ and $B$ be nonempty bounded subsets of $R$ such that $\sup A < \sup B$. Prove that there exists $e > 0$ and $b \in B$ so that $b - e$ if an upper bound of $A$.
I'm having trouble formalizing a proof for this.
Since A and B are bounded, by AoC we can say there is a supremum for both of them. We know $\sup A < \sup B$, which is saying that A has a smaller least upper bound than B. To prove that there is a positive $e$ that we can show $b-e$ is an upper bound of A for is what I am struggling to understand and how to prove.