3. SOME INEQUALITIES

25

Corollary

2165

.

1

.

|

R

|

= Compl(

|

R

|u ≥

);

2

.

|

R

|

>

= Compl(

|

R

|u

>

);

3

.

|

R

|

= Compl(

|

R

|u ≤

);

4

.

|

R

|

<

= Compl(

|

R

|u

<

).

Obvious

2166

.

FiXme

: also what is the values of

\

operation

1

.

|

R

|

=

|

R

|

>

t

1;

2

.

|

R

|

=

|

R

|

<

t

1.

3. Some inequalities

FiXme

: Define the ultrafilter “at the left” and “at the right” of a real number.

Also define “convergent ultrafilter”.

Denote ∆

+

=

d

x

R

]

x

; +

[ and ∆

−∞

=

d

x

R

]

− ∞

;

x

[.

The following proposition calculates

h≥i

x

and

h

>

i

x

for all kinds of ultrafilters

on

R

:

Proposition

2167

.

1

.

h≥i{

α

}

= [

α

; +

[ and

h

>

i{

α

}

=]

α

; +

[.

2

.

h≥i

x

=

h

>

i

x

=]

α

; +

[ for ultrafilter

x

at the right of a number

α

.

3

.

h≥i

x

=

h

>

i

x

= ∆

<

(

α

)

t

[

α

; +

[= ∆

(

α

)

t

]

α

; +

[ for ultrafilter

x

at the

left of a number

α

.

4

.

h≥i

x

=

h

>

i

x

= ∆

+

for ultrafilter

x

at positive infinity.

5

.

h≥i

x

=

h

>

i

x

=

R

for ultrafilter

x

at negative infinity.

Proof.

1

Obvious.

2

.

h≥i

x

=

F

l

X

up

x

h≥i

(

X

u

]

α

; +

[) =

F

l

X

up

x

]

α

; +

[=]

α

; +

[;

h

>

i

x

=

F

l

X

up

x

h

>

i

(

X

u

]

α

; +

[) =

F

l

X

up

x

]

α

; +

[=]

α

; +

[

.

3

<

(

α

)

t

[

α

; +

[= ∆

(

α

)

t

]

α

; +

[ is obvious.

h

>

i

x

=

F

l

X

up

x

h

>

i

X

w

F

l

X

up

x

(∆

<

(

α

)

t

]

α

; +

[) = ∆

<

(

α

)

t

]

α

; +

[

but

h≥i

x

v

<

(

α

)

t

[

α

; +

[ is obvious. It remains to take into account that

h

>

i

x

v h≥i

x

.

4

.

h≥i

x

=

d

F

X

up

x

h≥i

X

=

d

F

X

up

x,

inf

X

X

h≥i

(

X

u

]

α

; +

[)

=

d

F

X

up

x

[inf

X

; +

[=

d

F

x>α

[

x

; +

[=

+

;

h

>

i

x

=

d

F

X

up

x

h

>

i

X

=

d

F

X

up

x,

inf

X

X

h

>

i

(

X

u

]

α

; +

[) =

d

F

X

up

x

] inf

X

; +

[=

d

F

x>α

[

x

; +

[= ∆

+

.

5

.

h≥i

x

w h

>

i

x

=

d

F

X

up

x

h

>

i

X

but

h

>

i

X

=]

− ∞

; +

[ for

X

up

x

because

X

has arbitrarily small elements.

Lemma

2168

.

h|

R

|i

x

v h

>

i

x

=

h≥i

x

for every nontrivial ultrafilter

x

.

Proof.

h

>

i

x

=

h≥i

x

follows from the previous proposition.

h|

R

|i

x

=

d

X

up

x

h|

R

|i

X

=

d

X

up

x

d

y

X

∆(

y

).

Consider cases: