Characterizing the Supremum of a Set

Prove that if $S$ is an upper bound for a set $A$, then $S$ is the least upper bound (supremum) of $A$ if and only if for all $epsilon > 0$, there exists an a  in A such that $S - epsilon < a$.

In real analysis, one of the critical ideas is understanding the behavior and properties of bounds and supremums. This problem is centered around proving a fundamental characterization of the supremum or least upper bound of a set. Basically, a number is the supremum if it is the smallest possible upper bound — no other upper bounds can be smaller. However, for a number to be the supremum, it must also be intimately connected to the elements of the set it bounds. The problem is asking us to show that if S is an upper bound for a set A, it is the least upper bound if and only if for every positive quantity represented by epsilon, you can find an element in the set that is greater than S minus epsilon. This condition is crucial because it essentially squeezes S, showing that no number less than S can be an upper bound. Conceptually, understanding this property requires a grasp of how upper bounds relate to the elements of a set and the meaning of being "least." It involves epsilon arguments, which are common in analysis for proving properties related to bounds and limits. Working through such problems strengthens comprehension of the relationship between a set, its bounds, and the supremum. Recognizing that the slightest decrease from S still allows set elements to overtake it is key to appreciating the minimal and bounding nature of supremum. This idea is not only central to real analysis but also evolves into more complex scenarios in different mathematical contexts.