Definition:Bounded Above Set/Real Numbers
This page is about Bounded Above Subset of Real Numbers. For other uses, see Bounded Above.
Definition
Let $\R$ be the set of real numbers.
A subset $T \subseteq \R$ is bounded above (in $\R$) if and only if $T$ admits an upper bound (in $\R$).
Unbounded Above
$T \subseteq \R$ is unbounded above (in $\R$) if and only if it is not bounded above.
Examples
Example: $\hointl \gets 2$
The subset $I$ of the real numbers $\R$ defined as:
- $I = \hointl \gets 2$
is bounded above by, for example, $2$, $3$ and $4$, of which the supremum is $2$.
$2$ is also the greatest element of $I$.
The set of all upper bounds of $I$ is:
- $\closedint 2 \to$
Example: $\openint 0 1$
Let $I$ be the open real interval defined as:
- $I := \openint 0 1$
Then $I$ is bounded above by, for example, $1$, $2$ and $3$, of which $1$ is the supremum.
However, $I$ does not have a greatest element.
Example: $\set {-1, 0, 2, 5}$
Let $I$ be the set defined as:
- $I := \set {-1, 0, 2, 5}$
Then $I$ is bounded above by, for example, $5$, $6$ and $7$, of which the supremum is $5$.
$5$ is also the greatest element of $I$.
Example: $\openint 3 \to$
Let $I$ be the unbounded open real interval defined as:
- $I := \openint 3 \to$
Then $I$ is not bounded above.
Hence $I$ does not admit a supremum, and so does not have a greatest element.
Example: $\closedint 0 1$
Let $I$ be the closed real interval defined as:
- $I := \closedint 0 1$
Then $I$ is bounded above by, for example, $1$, $2$ and $3$, of which $1$ is the supremum.
$I$ is also the greatest element of $I$.
Also see
- Results about bounded above sets of real numbers can be found here.
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 ... (next): Part $\text {II}$: Foundations: Chapter $8$: Least Upper Bounds
- 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