A Supremum

Lingual

Mathematics

Calculus

The supremum (often abbreviated as sup) of a set of numbers is its least upper bound.

Every number in the set is less than or equal to the supremum, and no smaller number can act as an upper bound.

Supremum vs. Maximum

The most important distinction in real analysis is the difference between a maximum and a supremum.

• A maximum must actually be a member of the set.
• A supremum acts as a boundary; it does not have to be inside the set.

Formal Definition

Let \(S\) be a non-empty subset of real numbers (\(\mathbb{R}\)). A real number \(u\) is the supremum of \(S\), written as \(u = \sup(S)\), if it satisfies two conditions:

1. It is an upper bound: For every \(x \in S\), \(x \le u\).
2. It is the least upper bound: If \(v\) is any other upper bound of \(S\), then \(u \le v\).

Why is this concept necessary?

The supremum is the cornerstone of calculus and real analysis because it plugs the "holes" in the number line.

This is formally known as the Completeness Axiom of Real Numbers, which states that every non-empty set of real numbers that has an upper bound is guaranteed to have a supremum. This property is what separates the smooth, continuous real number line from the "holey" rational numbers (fractions), allowing us to rigorously define limits, derivatives, and integrals.