# Bound

Pronunciation: /baʊnd/ Explain

A lower bound of a set of numbers is a number such that the number is less than or equal to all numbers in the set.[2] An upper bound of a set of numbers is a number such that the number is greater than or equal to all numbers in the set.

The least upper bound of a set of numbers is the smallest number that is greater than or equal to all numbers in the set. The least upper bound is also called the supremum. A set is said to be bounded from above if it has a least upper bound.

The greatest lower bound of a set of numbers is the largest number that is less than or equal to all numbers in the set. The greatest lower bound is called the infimum. A set is said to be bounded from below if it has a greatest lower bound.

A set is bounded if it has an upper bound and a lower bound. A set is unbounded if it does not have an upper bound or does not have a lower bound.

### Examples

SetGreatest Lower BoundLeast Upper BoundIs Bounded?
All real numbers {x: x∈ℝ} None None No
All positive integers {x: x+ℤ} 1 None No
{x: x∈ℝ, -1 < x < 5} -1 5 Yes
{x: x∈ℝ, -1 ≤ x ≤ 5} -1 5 Yes

### Functions

When one says a function is bounded, one is referring to the range of the function. If the range of the function has a greatest lower bound, then one says that the function has a greatest lower bound.

### References

1. McAdams, David E.. All Math Words Dictionary, bound. 2nd Classroom edition 20150108-4799968. pg 27. Life is a Story Problem LLC. January 8, 2015. Buy the book
2. Veblen, Oswald and Lennes, N. J. . Introduction to Infinitesimal Analysis; Functions of One Real Variable. 1st edition. pp 3-7. www.archive.org. John Wiley and Sons. 1907. Last Accessed 6/19/2018. http://www.archive.org/stream/cu31924059316103#page/n18/mode/1up/search/bound. Buy the book
3. Phillips, E. G.. Functions of a Complex Variable. 8th edition. pp 8-9. www.archive.org. Oliver and Boyd. 1961. Last Accessed 6/19/2018. http://www.archive.org/stream/functionsofacomp029605mbp#page/n17/mode/1up/search/conjugate. Buy the book

• McAdams, David E.. Set. allmathwords.org. All Math Words Encyclopedia. Life is a Story Problem LLC. 6/22/2018. https://www.allmathwords.org/en/s/set.html.

McAdams, David E. Bound. 4/12/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. https://www.allmathwords.org/en/b/bound.html.

### Revision History

4/12/2019: Changed equations and expressions to new format. (McAdams, David E.)
1/29/2019: Fixed formatting. (McAdams, David E.)
12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)
6/22/2018: Removed broken links, updated license, implemented new markup. (McAdams, David E.)
10/2/2010: Initial version. (McAdams, David E.)

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