Closed Sets and Limits of Sequences

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Section Closed Sets and Limits of Sequences

Suppose we consider a sequence

(a_{n})

in a subset

A

of a metric space

X

that converges to a point

x .

Must it be the case that

x \in A ?

We consider this question in the next activity.

Activity 10.11.

Let

A = (0, 1)

and

B = [0, 1]

in

(R, d_{E}) .

For each positive integer

n,

let

a_{n} = \frac{1}{n} .

Note that the sequence

(a_{n})

is contained in both sets

A

and

B .

(a)

To what does the sequence

(a_{n})

converge in

R ?

(b)

Is

lim a_{n}

in

A ?

(c)

Is

lim a_{n} \in B ?

(d)

Name two significant differences between the sets

A

and

B

that account for the different responses in parts (b) and

(c) ?

Respond using the terminology we have introduced in this section.

The result of Activity 10.11 is encapsulated in the next theorem.

Theorem 10.15.

A subset

C

of a metric space

X

is closed if and only if whenever

(c_{n})

is a sequence in

C

that converges to a point

c \in X,

then

c \in C .

Proof.

Let

X

be a metric space and

C

a subset of

X .

First assume that

C

is closed. Let

(c_{n})

be a convergent sequence in

C

with

c = lim c_{n} .

So either

c \in C

or

c

is a limit point of

C .

Since

C

contains its limit points, either case gives us

c \in C .

So

lim c_{n} \in C .

The proof of the remaining implication is left to the next activity.

Activity 10.12.

Let

X

be a metric space and

C

a subset of

X .

In this activity we will prove that if any time a sequence

(c_{n})

in

C

converges to a point

c \in X,

the point

c

is in

C,

then

C

is a closed set.

(a)

List three different ways that we can show that a subset of a metric space is closed. Which one might be relevant in this situation to show that the set

C

is closed?

(b)

Let

c

be a limit point of

C .

What does that tell us?

(c)

Complete the proof that

C

is a closed set.

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