Continuum Property/Corollary

Theorem

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 known as the greatest lower bound property of the real numbers.

Proof

Let $T = \set {x \in \R: -x \in S}$.

By Set of Real Numbers is Bounded Below iff Set of Negatives is Bounded Above:

$T$ is bounded above.

Thus by the Continuum Property, $T$ admits a supremum in $\R$.

From Negative of Supremum is Infimum of Negatives:

$\ds -\sup_{x \mathop \in T} x = \map {\inf_{x \mathop \in T} } {-x}$

That is, by definition of $T$:

$\ds -\sup_{x \mathop \in T} x = \inf_{x \mathop \in S} x$

and so $S$ admits an infimum in $\R$.

$\blacksquare$

Sources

• 1947: James M. Hyslop: Infinite Series (3rd ed.) ... (previous) ... (next): Chapter $\text I$: Functions and Limits: $\S 3$: Bounds of a Function: Theorem $\text{A}$
• 1970: Arne Broman: Introduction to Partial Differential Equations ... (previous) ... (next): Chapter $1$: Fourier Series: $1.1$ Basic Concepts: $1.1.2$ Definitions
• 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): $\S 2.5$: Limits
• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $1$: Review of some real analysis: $\S 1.1$: Real Numbers: Proposition $1.1.5$
• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $1$: Review of some real analysis: Exercise $1.5: 6$
• 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 2$: Continuum Property: $\S 2.4$: The Continuum Property