Continuum Property

Theorem

Let $S \subset \R$ be a non-empty subset of the set of real numbers such that $S$ is bounded above.

Then $S$ admits a supremum in $\R$.

This is sometimes called the least upper bound property of the real numbers.

Similarly, let $S \subset \R$ be a non-empty subset of the set of real numbers such that $S$ is bounded below.

Then $S$ admits an infimum in $\R$.

This is sometimes called the greatest lower bound property of the real numbers.

The two properties taken together are called the continuum property of $\R$, or the completeness axiom.

Proof

A direct consequence of Dedekind's Theorem.

$\blacksquare$

Note

Not to be confused with:

• The Continuum Hypothesis;
• The Continuity Property.

Sources

• James M. Hyslop: Infinite Series (1942): $\S 3$: Theorem $\text{A}$
• W.A. Sutherland: Introduction to Metric and Topological Spaces (1975): Axiom $1.1.4$
• K.G. Binmore: Mathematical Analysis: A Straightforward Approach (1977)... (previous)... (next): $\S 2.2$