Supremum & Completeness

Least upper bounds, infimum, and the axiom that separates the reals from the rationals.

The idea

The supremum of a set of real numbers is its least upper bound. An upper bound of $S \subseteq \mathbb{R}$ is a number at least as large as every element of $S$; anything larger than an upper bound is an upper bound too, so we single out the smallest. That smallest one is the supremum $\sup S$: the number $b$ such that no element of $S$ exceeds $b$, and no number smaller than $b$ is an upper bound — any candidate $b - \varepsilon$ below it is beaten by some element of $S$. The same definition applied to lower bounds gives the infimum $\inf S$, the greatest lower bound.

The definition does not require $\sup S$ to belong to $S$. When it does, it is also the maximum of $S$; when it does not, $S$ has a supremum but no largest element.

That the supremum exists at all cannot be proved from arithmetic. We take it as an axiom, the Completeness Axiom: every nonempty subset of $\mathbb{R}$ that is bounded above has a least upper bound, and that bound is a real number.

The rationals lack this property: a set of rationals can be bounded above while no rational number is the least of its upper bounds. Completeness is the reason analysis works over $\mathbb{R}$ rather than $\mathbb{Q}$.

Ways to work on it

• Walkthrough. Define the supremum, distinguish it from the maximum, and state the Completeness Axiom.
• Practice. Find the supremum of a bounded set.
• Hardest. Find an irrational supremum and see why completeness is required.

Not sure where to start? Take the ten-question placement test.