Supremum, Infimum - Set Theory, CSIR-NET Mathematical Sciences | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET PDF Download

 Page 1


The supremum and in?mum
We review the de?nition of the supremum and and in?mum and some of their
properties that we use in de?ning and analyzing the Riemann integral.
2.1. De?nition
First, we de?ne upper and lower bounds.
De?nition 2.1. A setA?R of realnumbers is bounded fromabove if there exists
a real number M ? R, called an upper bound of A, such that x = M for every
x ? A. Similarly, A is bounded from below if there exists m ?R, called a lower
bound of A, such that x = m for every x ? A. A set is bounded if it is bounded
both from above and below.
The supremum of a set is its least upper bound and the in?mum is its greatest
upper bound.
De?nition 2.2. Suppose that A ? R is a set of real numbers. If M ? R is an
upper bound of A such that M = M
'
for every upper bound M
'
of A, then M is
called the supremum of A, denoted M = supA. If m ?R is a lower bound of A
such that m= m
'
for every lower bound m
'
of A, then m is called the or in?mum
of A, denoted m = infA.
If A is not bounded from above, then we write supA = 8, and if A is not
bounded from below, we write infA =-8. If A =Ø is the empty set, then every
real number is both an upper and a lower bound of A, and we write supØ =-8,
infØ =8. We will only say the supremum or in?mum of a set exists if it is a ?nite
real number. For an indexed set A ={x
k
:k? J}, we often write
supA = sup
k?J
x
k
, infA = inf
k?J
x
k
.
Proposition 2.3. The supremum or in?mum of a set A is unique if it exists.
Moreover, if both exist, then infA= supA.
Page 2


The supremum and in?mum
We review the de?nition of the supremum and and in?mum and some of their
properties that we use in de?ning and analyzing the Riemann integral.
2.1. De?nition
First, we de?ne upper and lower bounds.
De?nition 2.1. A setA?R of realnumbers is bounded fromabove if there exists
a real number M ? R, called an upper bound of A, such that x = M for every
x ? A. Similarly, A is bounded from below if there exists m ?R, called a lower
bound of A, such that x = m for every x ? A. A set is bounded if it is bounded
both from above and below.
The supremum of a set is its least upper bound and the in?mum is its greatest
upper bound.
De?nition 2.2. Suppose that A ? R is a set of real numbers. If M ? R is an
upper bound of A such that M = M
'
for every upper bound M
'
of A, then M is
called the supremum of A, denoted M = supA. If m ?R is a lower bound of A
such that m= m
'
for every lower bound m
'
of A, then m is called the or in?mum
of A, denoted m = infA.
If A is not bounded from above, then we write supA = 8, and if A is not
bounded from below, we write infA =-8. If A =Ø is the empty set, then every
real number is both an upper and a lower bound of A, and we write supØ =-8,
infØ =8. We will only say the supremum or in?mum of a set exists if it is a ?nite
real number. For an indexed set A ={x
k
:k? J}, we often write
supA = sup
k?J
x
k
, infA = inf
k?J
x
k
.
Proposition 2.3. The supremum or in?mum of a set A is unique if it exists.
Moreover, if both exist, then infA= supA.
Proof. Suppose that M, M
'
are suprema of A. Then M = M
'
since M
'
is an
upper bound of A and M is a least upper bound; similarly, M
'
=M, so M =M
'
.
If m, m
'
are in?ma of A, then m= m
'
since m
'
is a lower bound of A and m is a
greatest lower bound; similarly, m
'
=m, so m =m
'
.
If infA and supA exist, then A is nonempty. Choose x? A, Then
infA=x= supA
since infA is a lower bound of A and supA is an upper bound. It follows that
infA= supA. 
If supA? A, then we also denote it by maxA and call it the maximum of A,
and if infA? A, then we also denote it by minA and call it the minimum of A.
Example 2.4. Let A ={1/n :n?N}. Then supA = 1 belongs to A, so maxA =
1. On the other hand, infA = 0 doesn’t belong to A and A has no minimum.
The following alternative characterization of the sup and inf is an immediate
consequence of the de?nition.
Proposition 2.5. If A ? R, then M = supA if and only if: (a) M is an upper
bound of A; (b) for every M
'
<M there exists x?A such that x >M
'
. Similarly,
m = infA if and only if: (a) m is a lower bound of A; (b) for every m
'
> m there
exists x? A such that x<m
'
.
Proof. SupposeM satis?es the conditions in the proposition. ThenM is anupper
bound and (b) implies that if M
'
< M, then M
'
is not an an upper bound, so
M = supA. Conversely, if M = supA, then M is an upper bound, and if M
'
<M
then M
'
is not an upper bound, so there exists x?A such that x>M
'
. The proof
for the in?mum is analogous. 
Wefrequentlyuseoneofthefollowingarguments: (a)IfM isanupperboundof
A, thenM = supA; (b) For everyo> 0, there existsx? A suchthatx > supA-o.
Similarly: (a) If m is an lower bound of A, then m = infA; (b) For every o > 0,
there exists x?A such that x < infA+o.
The completeness of the real numbers ensures the existence of suprema and
in?ma. In fact, the existence of suprema and in?ma is one way to de?ne the
completeness ofR.
Theorem 2.6. Every nonempty set of real numbers that is bounded from above
has a supremum, and every nonempty set of real numbers that is bounded from
below has an in?mum.
This theorem is the basis of many existence results in real analysis. For exam-
ple, once we show that a set is bounded from above, we can assert the existence of
a supremum without having to know its actual value.
2.2. Properties
If A?R and c?R, then we de?ne
cA ={y?R :y =cx for some x? A}.
Page 3


The supremum and in?mum
We review the de?nition of the supremum and and in?mum and some of their
properties that we use in de?ning and analyzing the Riemann integral.
2.1. De?nition
First, we de?ne upper and lower bounds.
De?nition 2.1. A setA?R of realnumbers is bounded fromabove if there exists
a real number M ? R, called an upper bound of A, such that x = M for every
x ? A. Similarly, A is bounded from below if there exists m ?R, called a lower
bound of A, such that x = m for every x ? A. A set is bounded if it is bounded
both from above and below.
The supremum of a set is its least upper bound and the in?mum is its greatest
upper bound.
De?nition 2.2. Suppose that A ? R is a set of real numbers. If M ? R is an
upper bound of A such that M = M
'
for every upper bound M
'
of A, then M is
called the supremum of A, denoted M = supA. If m ?R is a lower bound of A
such that m= m
'
for every lower bound m
'
of A, then m is called the or in?mum
of A, denoted m = infA.
If A is not bounded from above, then we write supA = 8, and if A is not
bounded from below, we write infA =-8. If A =Ø is the empty set, then every
real number is both an upper and a lower bound of A, and we write supØ =-8,
infØ =8. We will only say the supremum or in?mum of a set exists if it is a ?nite
real number. For an indexed set A ={x
k
:k? J}, we often write
supA = sup
k?J
x
k
, infA = inf
k?J
x
k
.
Proposition 2.3. The supremum or in?mum of a set A is unique if it exists.
Moreover, if both exist, then infA= supA.
Proof. Suppose that M, M
'
are suprema of A. Then M = M
'
since M
'
is an
upper bound of A and M is a least upper bound; similarly, M
'
=M, so M =M
'
.
If m, m
'
are in?ma of A, then m= m
'
since m
'
is a lower bound of A and m is a
greatest lower bound; similarly, m
'
=m, so m =m
'
.
If infA and supA exist, then A is nonempty. Choose x? A, Then
infA=x= supA
since infA is a lower bound of A and supA is an upper bound. It follows that
infA= supA. 
If supA? A, then we also denote it by maxA and call it the maximum of A,
and if infA? A, then we also denote it by minA and call it the minimum of A.
Example 2.4. Let A ={1/n :n?N}. Then supA = 1 belongs to A, so maxA =
1. On the other hand, infA = 0 doesn’t belong to A and A has no minimum.
The following alternative characterization of the sup and inf is an immediate
consequence of the de?nition.
Proposition 2.5. If A ? R, then M = supA if and only if: (a) M is an upper
bound of A; (b) for every M
'
<M there exists x?A such that x >M
'
. Similarly,
m = infA if and only if: (a) m is a lower bound of A; (b) for every m
'
> m there
exists x? A such that x<m
'
.
Proof. SupposeM satis?es the conditions in the proposition. ThenM is anupper
bound and (b) implies that if M
'
< M, then M
'
is not an an upper bound, so
M = supA. Conversely, if M = supA, then M is an upper bound, and if M
'
<M
then M
'
is not an upper bound, so there exists x?A such that x>M
'
. The proof
for the in?mum is analogous. 
Wefrequentlyuseoneofthefollowingarguments: (a)IfM isanupperboundof
A, thenM = supA; (b) For everyo> 0, there existsx? A suchthatx > supA-o.
Similarly: (a) If m is an lower bound of A, then m = infA; (b) For every o > 0,
there exists x?A such that x < infA+o.
The completeness of the real numbers ensures the existence of suprema and
in?ma. In fact, the existence of suprema and in?ma is one way to de?ne the
completeness ofR.
Theorem 2.6. Every nonempty set of real numbers that is bounded from above
has a supremum, and every nonempty set of real numbers that is bounded from
below has an in?mum.
This theorem is the basis of many existence results in real analysis. For exam-
ple, once we show that a set is bounded from above, we can assert the existence of
a supremum without having to know its actual value.
2.2. Properties
If A?R and c?R, then we de?ne
cA ={y?R :y =cx for some x? A}.
Proposition 2.7. If c= 0, then
supcA =csupA, infcA =cinfA.
If c< 0, then
supcA =cinfA, infcA =csupA.
Proof. The result is obvious if c = 0. If c > 0, then cx = M if and only if
x= M/c, which shows that M is an upper bound of cA if and only if M/c is an
upper bound of A, so supcA = csupA. If c < 0, then then cx= M if and only if
x=M/c, so M is an upper bound of cA if and only if M/c is a lower bound of A,
so supcA =cinfA. The remaining results follow similarly. 
Making a set smaller decreases its supremum and increases its in?mum.
Proposition 2.8. Suppose that A, B are subsets ofR such that A?B. If supA
and supB exist, then supA= supB, and if infA, infB exist, then infA= infB.
Proof. Since supB is an upper bound of B and A ? B, it follows that supB is
an upper bound of A, so supA= supB. The proof for the in?mum is similar, or
apply the result for the supremum to-A?-B. 
Proposition 2.9. Suppose that A, B are nonempty sets of real numbers such that
x=y for all x?A and y?B. Then supA= infB.
Proof. Fix y ? B. Since x= y for all x? A, it follows that y is an upper bound
of A, so y= supA. Hence, supA is a lower bound of B, so supA= infB. 
If A,B?R are nonempty, we de?ne
A+B ={z :z =x+y for some x?A, y?B},
A-B ={z :z =x-y for some x?A, y?B}
Proposition 2.10. If A, B are nonempty sets, then
sup(A+B) = supA+supB, inf(A+B) = infA+infB,
sup(A-B) = supA-infB, inf(A-B) = infA-supB.
Proof. The set A+B is bounded from above if and only if A and B are bounded
from above, so sup(A+B) exists if and only if both supA and supB exist. In that
case, if x?A and y?B, then
x+y= supA+supB,
so supA+supB is an upper bound of A+B and therefore
sup(A+B)= supA+supB.
To get the inequality in the opposite direction, suppose that o > 0. Then there
exists x? A and y?B such that
x > supA-
o
2
, y > supB-
o
2
.
It follows that
x+y > supA+supB-o
Page 4


The supremum and in?mum
We review the de?nition of the supremum and and in?mum and some of their
properties that we use in de?ning and analyzing the Riemann integral.
2.1. De?nition
First, we de?ne upper and lower bounds.
De?nition 2.1. A setA?R of realnumbers is bounded fromabove if there exists
a real number M ? R, called an upper bound of A, such that x = M for every
x ? A. Similarly, A is bounded from below if there exists m ?R, called a lower
bound of A, such that x = m for every x ? A. A set is bounded if it is bounded
both from above and below.
The supremum of a set is its least upper bound and the in?mum is its greatest
upper bound.
De?nition 2.2. Suppose that A ? R is a set of real numbers. If M ? R is an
upper bound of A such that M = M
'
for every upper bound M
'
of A, then M is
called the supremum of A, denoted M = supA. If m ?R is a lower bound of A
such that m= m
'
for every lower bound m
'
of A, then m is called the or in?mum
of A, denoted m = infA.
If A is not bounded from above, then we write supA = 8, and if A is not
bounded from below, we write infA =-8. If A =Ø is the empty set, then every
real number is both an upper and a lower bound of A, and we write supØ =-8,
infØ =8. We will only say the supremum or in?mum of a set exists if it is a ?nite
real number. For an indexed set A ={x
k
:k? J}, we often write
supA = sup
k?J
x
k
, infA = inf
k?J
x
k
.
Proposition 2.3. The supremum or in?mum of a set A is unique if it exists.
Moreover, if both exist, then infA= supA.
Proof. Suppose that M, M
'
are suprema of A. Then M = M
'
since M
'
is an
upper bound of A and M is a least upper bound; similarly, M
'
=M, so M =M
'
.
If m, m
'
are in?ma of A, then m= m
'
since m
'
is a lower bound of A and m is a
greatest lower bound; similarly, m
'
=m, so m =m
'
.
If infA and supA exist, then A is nonempty. Choose x? A, Then
infA=x= supA
since infA is a lower bound of A and supA is an upper bound. It follows that
infA= supA. 
If supA? A, then we also denote it by maxA and call it the maximum of A,
and if infA? A, then we also denote it by minA and call it the minimum of A.
Example 2.4. Let A ={1/n :n?N}. Then supA = 1 belongs to A, so maxA =
1. On the other hand, infA = 0 doesn’t belong to A and A has no minimum.
The following alternative characterization of the sup and inf is an immediate
consequence of the de?nition.
Proposition 2.5. If A ? R, then M = supA if and only if: (a) M is an upper
bound of A; (b) for every M
'
<M there exists x?A such that x >M
'
. Similarly,
m = infA if and only if: (a) m is a lower bound of A; (b) for every m
'
> m there
exists x? A such that x<m
'
.
Proof. SupposeM satis?es the conditions in the proposition. ThenM is anupper
bound and (b) implies that if M
'
< M, then M
'
is not an an upper bound, so
M = supA. Conversely, if M = supA, then M is an upper bound, and if M
'
<M
then M
'
is not an upper bound, so there exists x?A such that x>M
'
. The proof
for the in?mum is analogous. 
Wefrequentlyuseoneofthefollowingarguments: (a)IfM isanupperboundof
A, thenM = supA; (b) For everyo> 0, there existsx? A suchthatx > supA-o.
Similarly: (a) If m is an lower bound of A, then m = infA; (b) For every o > 0,
there exists x?A such that x < infA+o.
The completeness of the real numbers ensures the existence of suprema and
in?ma. In fact, the existence of suprema and in?ma is one way to de?ne the
completeness ofR.
Theorem 2.6. Every nonempty set of real numbers that is bounded from above
has a supremum, and every nonempty set of real numbers that is bounded from
below has an in?mum.
This theorem is the basis of many existence results in real analysis. For exam-
ple, once we show that a set is bounded from above, we can assert the existence of
a supremum without having to know its actual value.
2.2. Properties
If A?R and c?R, then we de?ne
cA ={y?R :y =cx for some x? A}.
Proposition 2.7. If c= 0, then
supcA =csupA, infcA =cinfA.
If c< 0, then
supcA =cinfA, infcA =csupA.
Proof. The result is obvious if c = 0. If c > 0, then cx = M if and only if
x= M/c, which shows that M is an upper bound of cA if and only if M/c is an
upper bound of A, so supcA = csupA. If c < 0, then then cx= M if and only if
x=M/c, so M is an upper bound of cA if and only if M/c is a lower bound of A,
so supcA =cinfA. The remaining results follow similarly. 
Making a set smaller decreases its supremum and increases its in?mum.
Proposition 2.8. Suppose that A, B are subsets ofR such that A?B. If supA
and supB exist, then supA= supB, and if infA, infB exist, then infA= infB.
Proof. Since supB is an upper bound of B and A ? B, it follows that supB is
an upper bound of A, so supA= supB. The proof for the in?mum is similar, or
apply the result for the supremum to-A?-B. 
Proposition 2.9. Suppose that A, B are nonempty sets of real numbers such that
x=y for all x?A and y?B. Then supA= infB.
Proof. Fix y ? B. Since x= y for all x? A, it follows that y is an upper bound
of A, so y= supA. Hence, supA is a lower bound of B, so supA= infB. 
If A,B?R are nonempty, we de?ne
A+B ={z :z =x+y for some x?A, y?B},
A-B ={z :z =x-y for some x?A, y?B}
Proposition 2.10. If A, B are nonempty sets, then
sup(A+B) = supA+supB, inf(A+B) = infA+infB,
sup(A-B) = supA-infB, inf(A-B) = infA-supB.
Proof. The set A+B is bounded from above if and only if A and B are bounded
from above, so sup(A+B) exists if and only if both supA and supB exist. In that
case, if x?A and y?B, then
x+y= supA+supB,
so supA+supB is an upper bound of A+B and therefore
sup(A+B)= supA+supB.
To get the inequality in the opposite direction, suppose that o > 0. Then there
exists x? A and y?B such that
x > supA-
o
2
, y > supB-
o
2
.
It follows that
x+y > supA+supB-o
foreveryo> 0,whichimpliesthatsup(A+B)= supA+supB. Thus,sup(A+B) =
supA+supB.
It follows from this result and Proposition 2.7 that
sup(A-B) = supA+sup(-B) = supA-infB.
The proof of the results for inf(A +B) and inf(A-B) are similar, or apply the
results for the supremum to-A and-B. 
2.3. Functions
The supremum and in?mum of a function are the supremum and in?mum of its
range, and results about sets translate immediately to results about functions.
De?nition 2.11. If f :A?R is a function, then
sup
A
f = sup{f(x) :x?A}, inf
A
f = inf{f(x) :x?A}.
A function f is bounded from above on A if sup
A
f is ?nite, bounded from below
on A if inf
A
f is ?nite, and bounded on A if both are ?nite.
Inequalities and operations on functions are de?ned pointwise as usual; for
example, if f,g : A?R, then f =g means that f(x)= g(x) for every x? A, and
f +g :A?R is de?ned by (f +g)(x) =f(x)+g(x).
Proposition 2.12. Suppose that f,g : A ?R and f = g. If g is bounded from
above then
sup
A
f = sup
A
g,
and if f is bounded from below, then
inf
A
f = inf
A
g.
Proof. If f =g and g is bounded from above, then for every x?A
f(x)=g(x)= sup
A
g.
Thus, f is bounded from above by sup
A
g, so sup
A
f = sup
A
g. Similarly, g is
bounded from below by inf
A
f, so inf
A
g= inf
A
f. 
Note that f = g does not imply that sup
A
f = inf
A
g; to get that conclusion,
we need to know that f(x)=g(y) for all x,y?A and use Proposition 2.10.
Example 2.13. De?ne f,g : [0,1]?R by f(x) = 2x, g(x) = 2x+1. Then f < g
and
sup
[0,1]
f = 2, inf
[0,1]
f = 0, sup
[0,1]
g = 3, inf
[0,1]
g = 1.
Thus, sup
[0,1]
f > inf
[0,1]
g.
Like limits, the supremum and in?mum do not preserve strict inequalities in
general.
Page 5


The supremum and in?mum
We review the de?nition of the supremum and and in?mum and some of their
properties that we use in de?ning and analyzing the Riemann integral.
2.1. De?nition
First, we de?ne upper and lower bounds.
De?nition 2.1. A setA?R of realnumbers is bounded fromabove if there exists
a real number M ? R, called an upper bound of A, such that x = M for every
x ? A. Similarly, A is bounded from below if there exists m ?R, called a lower
bound of A, such that x = m for every x ? A. A set is bounded if it is bounded
both from above and below.
The supremum of a set is its least upper bound and the in?mum is its greatest
upper bound.
De?nition 2.2. Suppose that A ? R is a set of real numbers. If M ? R is an
upper bound of A such that M = M
'
for every upper bound M
'
of A, then M is
called the supremum of A, denoted M = supA. If m ?R is a lower bound of A
such that m= m
'
for every lower bound m
'
of A, then m is called the or in?mum
of A, denoted m = infA.
If A is not bounded from above, then we write supA = 8, and if A is not
bounded from below, we write infA =-8. If A =Ø is the empty set, then every
real number is both an upper and a lower bound of A, and we write supØ =-8,
infØ =8. We will only say the supremum or in?mum of a set exists if it is a ?nite
real number. For an indexed set A ={x
k
:k? J}, we often write
supA = sup
k?J
x
k
, infA = inf
k?J
x
k
.
Proposition 2.3. The supremum or in?mum of a set A is unique if it exists.
Moreover, if both exist, then infA= supA.
Proof. Suppose that M, M
'
are suprema of A. Then M = M
'
since M
'
is an
upper bound of A and M is a least upper bound; similarly, M
'
=M, so M =M
'
.
If m, m
'
are in?ma of A, then m= m
'
since m
'
is a lower bound of A and m is a
greatest lower bound; similarly, m
'
=m, so m =m
'
.
If infA and supA exist, then A is nonempty. Choose x? A, Then
infA=x= supA
since infA is a lower bound of A and supA is an upper bound. It follows that
infA= supA. 
If supA? A, then we also denote it by maxA and call it the maximum of A,
and if infA? A, then we also denote it by minA and call it the minimum of A.
Example 2.4. Let A ={1/n :n?N}. Then supA = 1 belongs to A, so maxA =
1. On the other hand, infA = 0 doesn’t belong to A and A has no minimum.
The following alternative characterization of the sup and inf is an immediate
consequence of the de?nition.
Proposition 2.5. If A ? R, then M = supA if and only if: (a) M is an upper
bound of A; (b) for every M
'
<M there exists x?A such that x >M
'
. Similarly,
m = infA if and only if: (a) m is a lower bound of A; (b) for every m
'
> m there
exists x? A such that x<m
'
.
Proof. SupposeM satis?es the conditions in the proposition. ThenM is anupper
bound and (b) implies that if M
'
< M, then M
'
is not an an upper bound, so
M = supA. Conversely, if M = supA, then M is an upper bound, and if M
'
<M
then M
'
is not an upper bound, so there exists x?A such that x>M
'
. The proof
for the in?mum is analogous. 
Wefrequentlyuseoneofthefollowingarguments: (a)IfM isanupperboundof
A, thenM = supA; (b) For everyo> 0, there existsx? A suchthatx > supA-o.
Similarly: (a) If m is an lower bound of A, then m = infA; (b) For every o > 0,
there exists x?A such that x < infA+o.
The completeness of the real numbers ensures the existence of suprema and
in?ma. In fact, the existence of suprema and in?ma is one way to de?ne the
completeness ofR.
Theorem 2.6. Every nonempty set of real numbers that is bounded from above
has a supremum, and every nonempty set of real numbers that is bounded from
below has an in?mum.
This theorem is the basis of many existence results in real analysis. For exam-
ple, once we show that a set is bounded from above, we can assert the existence of
a supremum without having to know its actual value.
2.2. Properties
If A?R and c?R, then we de?ne
cA ={y?R :y =cx for some x? A}.
Proposition 2.7. If c= 0, then
supcA =csupA, infcA =cinfA.
If c< 0, then
supcA =cinfA, infcA =csupA.
Proof. The result is obvious if c = 0. If c > 0, then cx = M if and only if
x= M/c, which shows that M is an upper bound of cA if and only if M/c is an
upper bound of A, so supcA = csupA. If c < 0, then then cx= M if and only if
x=M/c, so M is an upper bound of cA if and only if M/c is a lower bound of A,
so supcA =cinfA. The remaining results follow similarly. 
Making a set smaller decreases its supremum and increases its in?mum.
Proposition 2.8. Suppose that A, B are subsets ofR such that A?B. If supA
and supB exist, then supA= supB, and if infA, infB exist, then infA= infB.
Proof. Since supB is an upper bound of B and A ? B, it follows that supB is
an upper bound of A, so supA= supB. The proof for the in?mum is similar, or
apply the result for the supremum to-A?-B. 
Proposition 2.9. Suppose that A, B are nonempty sets of real numbers such that
x=y for all x?A and y?B. Then supA= infB.
Proof. Fix y ? B. Since x= y for all x? A, it follows that y is an upper bound
of A, so y= supA. Hence, supA is a lower bound of B, so supA= infB. 
If A,B?R are nonempty, we de?ne
A+B ={z :z =x+y for some x?A, y?B},
A-B ={z :z =x-y for some x?A, y?B}
Proposition 2.10. If A, B are nonempty sets, then
sup(A+B) = supA+supB, inf(A+B) = infA+infB,
sup(A-B) = supA-infB, inf(A-B) = infA-supB.
Proof. The set A+B is bounded from above if and only if A and B are bounded
from above, so sup(A+B) exists if and only if both supA and supB exist. In that
case, if x?A and y?B, then
x+y= supA+supB,
so supA+supB is an upper bound of A+B and therefore
sup(A+B)= supA+supB.
To get the inequality in the opposite direction, suppose that o > 0. Then there
exists x? A and y?B such that
x > supA-
o
2
, y > supB-
o
2
.
It follows that
x+y > supA+supB-o
foreveryo> 0,whichimpliesthatsup(A+B)= supA+supB. Thus,sup(A+B) =
supA+supB.
It follows from this result and Proposition 2.7 that
sup(A-B) = supA+sup(-B) = supA-infB.
The proof of the results for inf(A +B) and inf(A-B) are similar, or apply the
results for the supremum to-A and-B. 
2.3. Functions
The supremum and in?mum of a function are the supremum and in?mum of its
range, and results about sets translate immediately to results about functions.
De?nition 2.11. If f :A?R is a function, then
sup
A
f = sup{f(x) :x?A}, inf
A
f = inf{f(x) :x?A}.
A function f is bounded from above on A if sup
A
f is ?nite, bounded from below
on A if inf
A
f is ?nite, and bounded on A if both are ?nite.
Inequalities and operations on functions are de?ned pointwise as usual; for
example, if f,g : A?R, then f =g means that f(x)= g(x) for every x? A, and
f +g :A?R is de?ned by (f +g)(x) =f(x)+g(x).
Proposition 2.12. Suppose that f,g : A ?R and f = g. If g is bounded from
above then
sup
A
f = sup
A
g,
and if f is bounded from below, then
inf
A
f = inf
A
g.
Proof. If f =g and g is bounded from above, then for every x?A
f(x)=g(x)= sup
A
g.
Thus, f is bounded from above by sup
A
g, so sup
A
f = sup
A
g. Similarly, g is
bounded from below by inf
A
f, so inf
A
g= inf
A
f. 
Note that f = g does not imply that sup
A
f = inf
A
g; to get that conclusion,
we need to know that f(x)=g(y) for all x,y?A and use Proposition 2.10.
Example 2.13. De?ne f,g : [0,1]?R by f(x) = 2x, g(x) = 2x+1. Then f < g
and
sup
[0,1]
f = 2, inf
[0,1]
f = 0, sup
[0,1]
g = 3, inf
[0,1]
g = 1.
Thus, sup
[0,1]
f > inf
[0,1]
g.
Like limits, the supremum and in?mum do not preserve strict inequalities in
general.
Example 2.14. De?ne f : [0,1]?R by
f(x) =
(
x if 0=x< 1,
0 if x = 1.
Then f < 1 on [0,1] but sup
[0,1]
f = 1.
Next, we consider the supremum and in?mum of linear combinations of func-
tions. Scalar multiplication by a positive constant multiplies the inf or sup, while
multiplication by a negative constant switches the inf and sup,
Proposition 2.15. Suppose that f : A?R is a bounded function and c?R. If
c= 0, then
sup
A
cf =csup
A
f, inf
A
cf =cinf
A
f.
If c< 0, then
sup
A
cf =cinf
A
f, inf
A
cf =csup
A
f.
Proof. Apply Proposition 2.7 to the set{cf(x) : x?A} =c{f(x) :x?A}. 
For sums of functions, we get an inequality.
Proposition 2.16. If f,g :A?R are bounded functions, then
sup
A
(f +g)= sup
A
f +sup
A
g, inf
A
(f +g)= inf
A
f +inf
A
g.
Proof. Since f(x)= sup
A
f and g(x)= sup
A
g for evry x? [a,b], we have
f(x)+g(x)= sup
A
f +sup
A
g.
Thus, f +g is bounded from above by sup
A
f +sup
A
g, so sup
A
(f +g)= sup
A
f +
sup
A
g. The proof for the in?mum is analogous (or apply the result for the supre-
mum to the functions-f,-g). 
We may have strict inequality in Proposition 2.16 because f and g may take
values close to their suprema (or in?ma) at di?erent points.
Example 2.17. De?ne f,g : [0,1]?R by f(x) = x, g(x) = 1-x. Then
sup
[0,1]
f = sup
[0,1]
g = sup
[0,1]
(f +g) = 1,
so sup
[0,1]
(f +g)< sup
[0,1]
f +sup
[0,1]
g.
Finally, we prove some inequalities that involve the absolute value.
Proposition 2.18. If f,g :A?R are bounded functions, then




sup
A
f-sup
A
g




= sup
A
|f-g|,


inf
A
f-inf
A
g


= sup
A
|f-g|.