Supremum of a Bounded Set

Suppose S is a non-empty subset of the real numbers that is bounded above by M. Then S has a least upper bound, meaning the supremum exists.

This problem explores the fundamental principle that every non-empty set of real numbers that is bounded above must have a least upper bound, a concept known as the supremum. This principle is crucial in real analysis and relates closely to the Completeness Axiom, which asserts that the real numbers are complete with respect to this property. The existence of the supremum has wide-reaching implications across various fields of mathematics, notably in proofs and theorems relating to limits, continuity, and integration.

When approaching problems like this, it's important to understand the definitions and the properties that come into play. The terms 'bounded above' and 'least upper bound' are central here. A set is bounded above if there is a real number that is greater than or equal to every element in the set. The supremum is the smallest of these upper bounds. Conceptually, you can think of the supremum as the 'ceiling' of the set, the point beyond which no elements of the set can go, albeit it does not have to be an actual member of the set.

In terms of strategy, proving the existence of the supremum often involves demonstrating that any proposed bound can be approached as closely as desired by members of the set. This approach can often be employed in conjunction with the Completeness Axiom or by constructing sequences that converge towards the supremum. Understanding these methods provides a deeper insight into how mathematicians guarantee the completeness property of the real number line. This problem thus serves as a foundation for subsequent concepts and proofs in the study of real analysis.