Definition:Lower Bound of Set/Real Numbers

This page is about Lower Bound of Subset of Real Numbers. For other uses, see Lower Bound.

Definition

Let $\R$ be the set of real numbers.

Let $T$ be a subset of $S$.

A lower bound for $T$ (in $\R$) is an element $m \in \R$ such that:

$\forall t \in T: m \le t$

Also see

• Definition:Bounded Below Subset of Real Numbers

• Definition:Upper Bound of Subset of Real Numbers
• Definition:Bounded Above Subset of Real Numbers

• Definition:Bounded Subset of Real Numbers

• Definition:Supremum of Subset of Real Numbers
• Definition:Infimum of Subset of Real Numbers

Sources

• 1964: Walter Rudin: Principles of Mathematical Analysis (2nd ed.) ... (previous) ... (next): Chapter $1$: The Real and Complex Number Systems: Real Numbers: $1.33$. Definition
• 1967: Michael Spivak: Calculus ... (previous) ... (next): Part $\text {II}$: Foundations: Chapter $8$: Least Upper Bounds
• 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $2$: Metric Spaces: $\S 5$: Limits: Definition $5.5$
• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $1$: Review of some real analysis: $\S 1.1$: Real Numbers
• 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 2$: Continuum Property: $\S 2.2$: The Continuum Property
• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 10$: The well-ordering principle
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): order properties (of real numbers)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): order properties (of real numbers)
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): bound