I am confused on this:
Suppose A is a non-empty set of $\mathbb{R}$ with finite supremum sup A = L. Then how do I prove that $\forall \epsilon>0, \exists a \in A $ s.t $L-a < \epsilon$. Maybe I can assume that $L-a>\epsilon$ or $L-a=\epsilon$ for a contradiction, but I am not sure how to proceed. Intuitively it seems to say that the difference between the supremum and any element of the subset can be as small as we want. How should I prove this point?