infimum and supremum for real numbers

infimum and supremum for real numbers

Suppose

A

is a non-empty subset of

ℝ

.  If

A

is

bounded from above

, then
the axioms of the

real numbers

imply that there exists
a

least

upper bound

for

A

.
That is, there exists an

m

∈

ℝ

such that

1.

m

is an upper bound for

A

, that is,

a

≤

m

for all

a

∈

A

,

2.

if

M

is another upper bound for

A

, then

m

≤

M

.

Such a number

m

is called the

supremum

of

A

,
and it is denoted by

sup

⁡

A

. It is easy to see that
there can be only one least upper bound. If

m

1

and

m

2

are
two least upper bounds for

A

. Then

m

1

≤

m

2

and

m

2

≤

m

1

,
and

m

1

=

m

2

.

Next, let us consider a set

A

that is bounded from below. That is, for
some

m

∈

ℝ

we have

m

≤

a

for all

a

∈

A

.  Then we say that

M

∈

ℝ

is a
a

greatest lower bound

for

A

if

1.

M

is an

lower bound

for

A

, that is,

M

≤

a

for all

a

∈

A

,

2.

if

m

is another lower bound for

A

, then

m

≤

M

.

Such a number

M

is called the

infimum

of

A

,
and it is denoted by

inf

⁡

A

. Just as we proved that the supremum
is unique, one can also show that the infimum is unique.
The next lemma shows that the infimum exists.

Lemma 1

.

Every non-empty set bounded from below has a greatest lower bound.

Proof.

Let

m

∈

ℝ

be a lower bound for non-empty set

A

.
In other words,

m

≤

a

for all

a

∈

A

. Let

-

A

=

{

-

a

∈

ℝ

:

a

∈

A

}

.

Let us recall the following result from this page (

http://planetmath.org/InequalityForRealNumbers

);
if

m

is an upper(lower) bound for

A

, then

-

m

is a lower(upper)
bound for

-

A

.

Thus

-

A

is bounded from above by

-

m

.
It follows that

-

A

has a least upper bound

sup

⁡

(

-

A

)

.
Now

-

sup

⁡

(

-

A

)

is a greatest lower bound for

A

.
First, by the result, it is a lower bound for

A

.
Second, if

m

is a lower bound for

A

,
then

-

m

is a upper bound for

-

A

, and

sup

⁡

(

-

A

)

≤

-

m

, or

m

≥

-

sup

⁡

(

-

A

)

.
∎

The proof shows that if

A

is non-empty and bounded from below,
then

inf

⁡

A

=

-

sup

⁡

(

-

A

)

.

In

consequence

, if

A

is bounded from above,
then

sup

⁡

A

=

-

inf

⁡

(

-

A

)

.

In many respects, the supremum and infimum are

similar

to the

maximum

and

minimum

, or the largest and smallest

element

in a set.
However, it is important to notice that the

inf

⁡

A

and

sup

⁡

A

do not need to belong to

A

. (See examples below.)

Examples

1.

For example, consider the set of negative real numbers

A

=

{

x

∈

ℝ

:

x

<

0

}

.

Then

sup

⁡

A

=

0

. Indeed. First,

a

<

0

for all

a

∈

A

,
and if

a

<

b

for all

a

∈

A

, then

0

≤

b

.

2.

The

sequence

-

(

1

-

1

1

)

,

1

-

1

2

,

-

(

1

-

1

3

)

,

1

-

1

4

,

-

(

1

-

1

5

)

,

…

is not

convergent

.  The set

A

=

{

(

-

1

)

n

⁢

(

1

-

1

n

)

:

n

∈

ℤ

+

}

formed by its members has the infimum

-

1

and the supremum 1.

Title

infimum and supremum for real numbers

Canonical name

InfimumAndSupremumForRealNumbers

Date of creation

2013-03-22 15:41:42

Last modified on

2013-03-22 15:41:42

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

6

Author

matte (1858)

Entry type

Topic

Classification

msc 54C30

Classification

msc 26-00

Classification

msc 12D99

Related topic

SetsThatDoNotHaveAnInfimum

Related topic

Infimum

Related topic

Supremum

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