Continuum Property/Corollary
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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): Chapter $2$: Metric Spaces: $\S 5$: 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
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Entry: order properties (of real numbers)