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Characteristics of sets on the real line. The least upper bound of a given set of real numbers is the smallest number bounding this set from above; its greatest lower bound is the largest number bounding it from below. This will now be restated in more detail. Let there be given a subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095810/u0958101.png" /> of the real numbers. A number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095810/u0958102.png" /> is said to be its least upper bound, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095810/u0958103.png" /> (from the Latin  "supremum"  — largest), if every number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095810/u0958104.png" /> satisfies the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095810/u0958105.png" />, and if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095810/u0958106.png" /> there exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095810/u0958107.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095810/u0958108.png" />. A number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095810/u0958109.png" /> is said to be the greatest lower bound of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095810/u09581010.png" />, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095810/u09581011.png" /> (from the Latin  "infimum"  — smallest), if every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095810/u09581012.png" /> satisfies the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095810/u09581013.png" />, and if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095810/u09581014.png" /> there exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095810/u09581015.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095810/u09581016.png" />.
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Characteristics of sets on the real line. The least upper bound of a given set of real numbers is the smallest number bounding this set from above; its greatest lower bound is the largest number bounding it from below. This will now be restated in more detail. Let there be given a subset $X$ of the real numbers. A number $\beta$ is said to be its least upper bound, denoted by $\sup X$ (from the Latin  "supremum"  — largest), if every number $x\in X$ satisfies the inequality $x\leq\beta$, and if for any $\beta'<\beta$ there exists an $x'\in X$ such that $x'>\beta'$. A number $\alpha$ is said to be the greatest lower bound of $X$, denoted by $\inf X$ (from the Latin  "infimum"  — smallest), if every $x\in X$ satisfies the inequality $x\geq\alpha$, and if for any $\alpha'>\alpha$ there exists an $x'\in X$ such that $x'<\alpha'$.
  
 
Examples.
 
Examples.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095810/u09581017.png" /></td> </tr></table>
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$$\inf[a,b]=a,\quad\sup[a,b]=b;$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095810/u09581018.png" /></td> </tr></table>
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$$\inf(a,b)=a,\quad\sup(a,b)=b;$$
  
if the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095810/u09581019.png" /> consists of two points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095810/u09581020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095810/u09581021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095810/u09581022.png" />, then
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if the set $X$ consists of two points $a$ and $b$, $a<b$, then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095810/u09581023.png" /></td> </tr></table>
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$$\inf X=a,\quad\sup X=b.$$
  
These examples show, in particular, that the least upper bound (greatest lower bound) may either belong to the set (e.g. in the case of the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095810/u09581024.png" />) or not belong to it (e.g. in the case of the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095810/u09581025.png" />). If a set has a largest (smallest) member, this number will clearly be the least upper bound (greatest lower bound) of the set.
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These examples show, in particular, that the least upper bound (greatest lower bound) may either belong to the set (e.g. in the case of the interval $[a,b]$) or not belong to it (e.g. in the case of the interval $(a,b)$). If a set has a largest (smallest) member, this number will clearly be the least upper bound (greatest lower bound) of the set.
  
The least upper bound (greatest lower bound) of a set not bounded from above (from below) is denoted by the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095810/u09581026.png" /> (respectively, by the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095810/u09581027.png" />). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095810/u09581028.png" /> is the set of natural numbers, then
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The least upper bound (greatest lower bound) of a set not bounded from above (from below) is denoted by the symbol $+\infty$ (respectively, by the symbol $-\infty$). If $\mathbf N=\{1,2,\dots\}$ is the set of natural numbers, then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095810/u09581029.png" /></td> </tr></table>
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$$\inf\mathbf N=1,\quad\sup\mathbf N=+\infty.$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095810/u09581030.png" /> is the set of all integers, both positive and negative, then
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If $\mathbf Z$ is the set of all integers, both positive and negative, then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095810/u09581031.png" /></td> </tr></table>
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$$\inf\mathbf Z=-\infty,\quad\sup\mathbf Z=+\infty.$$
  
 
Each non-empty set of real numbers has a unique least upper bound (greatest lower bound), finite or infinite. All non-empty sets bounded from above have finite least upper bounds, while all those bounded from below have finite greatest lower bounds.
 
Each non-empty set of real numbers has a unique least upper bound (greatest lower bound), finite or infinite. All non-empty sets bounded from above have finite least upper bounds, while all those bounded from below have finite greatest lower bounds.
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====Comments====
 
====Comments====
Commonly, an upper bound of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095810/u09581032.png" /> of real numbers is a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095810/u09581033.png" /> such that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095810/u09581034.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095810/u09581035.png" />. The least upper bound of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095810/u09581036.png" /> is then defined as an upper bound <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095810/u09581037.png" /> such that for every upper bound <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095810/u09581038.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095810/u09581039.png" />.
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Commonly, an upper bound of a set $S$ of real numbers is a number $b$ such that for all $x\in S$ one has $x\leq b$. The least upper bound of $S$ is then defined as an upper bound $B$ such that for every upper bound $b$ one has $B\leq b$.
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Analogous definitions hold for a lower bound and the greatest lower bound. If the least upper bound of $S$ belongs to $S$, then it is called the maximum of $S$.
  
Analogous definitions hold for a lower bound and the greatest lower bound. If the least upper bound of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095810/u09581040.png" /> belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095810/u09581041.png" />, then it is called the maximum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095810/u09581042.png" />.
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If the greatest lower bound of $S$ belongs to $S$, then it is called the minimum of $S$.
  
If the greatest lower bound of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095810/u09581043.png" /> belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095810/u09581044.png" />, then it is called the minimum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095810/u09581045.png" />.
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The fundamental axiom of the [[real number]] system, or [[continuity axiom]], may be expressed in the form ''Every non-empty set of real numbers bounded above has a real number supremum''.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  T.M. Apostol,  "Mathematical analysis" , Addison-Wesley  (1974)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W. Rudin,  "Principles of mathematical analysis" , McGraw-Hill  (1953)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  K.R. Stromberg,  "Introduction to classical real analysis" , Wadsworth  (1981)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  T.M. Apostol,  "Mathematical analysis" , Addison-Wesley  (1974)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W. Rudin,  "Principles of mathematical analysis" , McGraw-Hill  (1953)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  K.R. Stromberg,  "Introduction to classical real analysis" , Wadsworth  (1981)</TD></TR></table>

Latest revision as of 19:45, 7 January 2015

2020 Mathematics Subject Classification: Primary: 26A03 Secondary: 06A [MSN][ZBL]

Characteristics of sets on the real line. The least upper bound of a given set of real numbers is the smallest number bounding this set from above; its greatest lower bound is the largest number bounding it from below. This will now be restated in more detail. Let there be given a subset $X$ of the real numbers. A number $\beta$ is said to be its least upper bound, denoted by $\sup X$ (from the Latin "supremum" — largest), if every number $x\in X$ satisfies the inequality $x\leq\beta$, and if for any $\beta'<\beta$ there exists an $x'\in X$ such that $x'>\beta'$. A number $\alpha$ is said to be the greatest lower bound of $X$, denoted by $\inf X$ (from the Latin "infimum" — smallest), if every $x\in X$ satisfies the inequality $x\geq\alpha$, and if for any $\alpha'>\alpha$ there exists an $x'\in X$ such that $x'<\alpha'$.

Examples.

$$\inf[a,b]=a,\quad\sup[a,b]=b;$$

$$\inf(a,b)=a,\quad\sup(a,b)=b;$$

if the set $X$ consists of two points $a$ and $b$, $a<b$, then

$$\inf X=a,\quad\sup X=b.$$

These examples show, in particular, that the least upper bound (greatest lower bound) may either belong to the set (e.g. in the case of the interval $[a,b]$) or not belong to it (e.g. in the case of the interval $(a,b)$). If a set has a largest (smallest) member, this number will clearly be the least upper bound (greatest lower bound) of the set.

The least upper bound (greatest lower bound) of a set not bounded from above (from below) is denoted by the symbol $+\infty$ (respectively, by the symbol $-\infty$). If $\mathbf N=\{1,2,\dots\}$ is the set of natural numbers, then

$$\inf\mathbf N=1,\quad\sup\mathbf N=+\infty.$$

If $\mathbf Z$ is the set of all integers, both positive and negative, then

$$\inf\mathbf Z=-\infty,\quad\sup\mathbf Z=+\infty.$$

Each non-empty set of real numbers has a unique least upper bound (greatest lower bound), finite or infinite. All non-empty sets bounded from above have finite least upper bounds, while all those bounded from below have finite greatest lower bounds.

The terms least upper (greatest lower) limit of a set are also sometimes used instead of the least upper bound (greatest lower bound) of a set, in one of the senses defined above. By the least upper bound (greatest lower bound) of a real-valued function, in particular of a sequence of real numbers, one means the least upper bound (greatest lower bound) of the set of its values (cf. also Upper and lower limits).

References

[1] V.A. Il'in, E.G. Poznyak, "Fundamentals of mathematical analysis" , 1–2 , MIR (1982) (Translated from Russian)
[2] L.D. Kudryavtsev, "A course in mathematical analysis" , 1 , Moscow (1988) (In Russian)
[3] S.M. Nikol'skii, "A course of mathematical analysis" , 1 , MIR (1977) (Translated from Russian)


Comments

Commonly, an upper bound of a set $S$ of real numbers is a number $b$ such that for all $x\in S$ one has $x\leq b$. The least upper bound of $S$ is then defined as an upper bound $B$ such that for every upper bound $b$ one has $B\leq b$.

Analogous definitions hold for a lower bound and the greatest lower bound. If the least upper bound of $S$ belongs to $S$, then it is called the maximum of $S$.

If the greatest lower bound of $S$ belongs to $S$, then it is called the minimum of $S$.

The fundamental axiom of the real number system, or continuity axiom, may be expressed in the form Every non-empty set of real numbers bounded above has a real number supremum.

References

[a1] T.M. Apostol, "Mathematical analysis" , Addison-Wesley (1974)
[a2] W. Rudin, "Principles of mathematical analysis" , McGraw-Hill (1953)
[a3] K.R. Stromberg, "Introduction to classical real analysis" , Wadsworth (1981)
How to Cite This Entry:
Upper and lower bounds. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Upper_and_lower_bounds&oldid=15559
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article