limit of sequence of sets

limit of sequence of sets

Recall that

lim sup

and

lim inf

of a

sequence

of sets

{

A

i

}

denote the and the of

{

A

i

}

, respectively. Please click here (

http://planetmath.org/LimitSuperiorOfSets

) to see the definitions and here (

http://planetmath.org/LimitSuperior

) to see the specialized definitions when they are applied to the

real numbers

.

Theorem

. Let

{

A

i

}

be a sequence of sets with

i

∈

ℤ

+

=

{

1

,

2

,

…

}

. Then

1.

for

I

ranging over all

infinite subsets

of

ℤ

+

,

lim sup

⁡

A

i

=

⋃

I

⋂

i

∈

I

A

i

,

2.

for

I

ranging over all subsets of

ℤ

+

with finite compliment,

lim inf

⁡

A

i

=

⋃

I

⋂

i

∈

I

A

i

,

3.

lim inf

⁡

A

i

⊆

lim sup

⁡

A

i

.

Proof

.

1.

We need to show, for

I

ranging over all infinite subsets of

ℤ

+

,

⋃

I

⋂

i

∈

I

A

i

=

⋂

n

=

1

∞

⋃

i

=

n

∞

A

k

.

(1)

Let

x

be an element of the
LHS, the left hand side of Equation (1). Then

x

∈

⋂

i

∈

I

A

i

for some infinite subset

I

⊆

ℤ

+

. Certainly,

x

∈

⋃

i

=

1

∞

A

i

. Now, suppose

x

∈

⋃

i

=

k

∞

A

i

. Since

I

is

infinite

, we can
find an

l

∈

I

such that

l

>

k

. Being a member of

I

, we have
that

x

∈

A

l

⊆

⋃

i

=

k

+

1

∞

A

i

. By

induction

,
we have

x

∈

⋃

i

=

n

∞

A

i

for all

n

∈

ℤ

+

. Thus

x

is an element of the RHS. This
proves one side of the

inclusion

(

⊆

) in (1).

To show the other inclusion, let

x

be an element of the RHS. So

x

∈

⋃

i

=

n

∞

A

i

for all

n

∈

ℤ

+

In

⋃

i

=

1

∞

A

i

, pick the least

element

n

0

such that

x

∈

A

n

0

. Next, in

⋃

i

=

n

0

+

1

∞

A

i

, pick the
least

n

1

such that

x

∈

A

n

1

. Then the set

I

=

{

n

0

,

n

1

,

…

}

fulfills the requirement

x

∈

⋂

i

∈

I

A

i

, showing the other inclusion (

⊇

).

2.

Here we have to show, for

I

ranging over all subsets of

ℤ

+

with

ℤ

+

-

I

finite,

⋃

I

⋂

i

∈

I

A

i

=

⋃

n

=

1

∞

⋂

i

=

n

∞

A

k

.

(2)

Suppose first that

x

is
an element of the LHS so that

x

∈

⋂

i

∈

I

A

i

for some

I

with

ℤ

+

-

I

finite. Let

n

0

be a

upper bound

of
the

finite set

ℤ

+

-

I

such that for any

n

∈

ℤ

+

-

I

,

n

<

n

0

. This means that any

m

≥

n

0

,
we have

m

∈

I

. Therefore,

x

∈

⋂

i

=

n

0

∞

A

i

and

x

is an element of the RHS.

Next, suppose

x

is an element of the RHS so that

x

∈

⋂

k

=

n

∞

A

k

for some

n

. Then the set

I

=

{

n

0

,

n

0

+

1

,

…

}

is a subset of

ℤ

+

with finite

complement

that does the job for the LHS.

3.

The set of all subsets (of

ℤ

+

) with finite
complement is a subset of the set of all infinite subsets. The
third assertion is now clear from the previous two

propositions

. QED

Corollary

. If

{

A

i

}

is a

decreasing
sequence

of sets, then

lim inf

⁡

A

i

=

lim sup

⁡

A

i

=

lim

⁡

A

i

=

⋂

A

i

.

Similarly, if

{

A

i

}

is an

increasing
sequence

of sets, then

lim inf

⁡

A

i

=

lim sup

⁡

A

i

=

lim

⁡

A

i

=

⋃

A

i

.

Proof

. We shall only show the case when we have a
descending chain of sets, since the other case is completely
analogous. Let

A

1

⊇

A

2

⊇

…

be a descending
chain of sets. Set

A

=

⋂

i

=

1

∞

A

i

. We shall show
that

lim sup

⁡

A

i

=

lim inf

⁡

A

i

=

lim

⁡

A

i

=

A

.

First, by the
definition of of a sequence of sets:

lim sup

⁡

A

i

=

⋂

n

=

1

∞

⋃

i

=

n

∞

A

k

=

⋂

n

=

1

∞

A

n

=

A

.

Now, by Assertion 3 of the above
Theorem,

lim inf

⁡

A

i

⊆

lim sup

⁡

A

i

=

A

, so we only need to
show that

A

⊆

lim inf

⁡

A

i

. But this is immediate from the
definition of

A

, being the

intersection

of all

A

i

with

subscripts

i

taking on all values of

ℤ

+

. Its complement is the

empty
set

, clearly finite. Having shown both the

existence

and equality
of the and of the

A

i

’s, we conclude
that the limit of

A

i

’s exist as well and it is equal to

A

.
QED

Title

limit of sequence of sets

Canonical name

LimitOfSequenceOfSets

Date of creation

2013-03-22 15:00:34

Last modified on

2013-03-22 15:00:34

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

8

Author

CWoo (3771)

Entry type

Theorem

Classification

msc 03E20

Classification

msc 28A05

Classification

msc 60A99

Related topic

LimitSuperior

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