Identifying Sequentially Compact Sets Quiz

1) A = {(-1)^n/n} sequentially compact because:

It is finite

Contains monotone seq

Bounded & closure contains 0

All seq constant

Sequence converges to 0.

Explanation

Sequence converges to 0.

2) In ℝ, every sequentially compact set must contain all its limit points.

True

False

Sequentially compact ⇔ compact ⇔ closed & bounded.

Explanation

Sequentially compact ⇔ compact ⇔ closed & bounded.

3) The set {1/n} is sequentially compact in ℝ.

True

False

Limit point 0 not in set.

Explanation

Limit point 0 not in set.

4) Any infinite subset of ℕ is sequentially compact.

True

False

ℕ unbounded, no convergent subsequence.

Explanation

ℕ unbounded, no convergent subsequence.

5) A finite subset of any metric space is always sequentially compact.

True

False

Constant subsequence exists.

Explanation

Constant subsequence exists.

6) Which subset of ℝ is sequentially compact?

(1,2]

[1,2)

[1,2]

(1,2)

Only [1,2] closed & bounded.

Explanation

Only [1,2] closed & bounded.

7) Which set is not sequentially compact?

[0,1]∪[2,3]

Finite set

[0,1)

{1+1/n}

[0,1) not closed.

Explanation

[0,1) not closed.

8) (0,1] ∪ {2−1/n} is:

Sequentially compact

Not seq compact unbounded

Not seq compact not closed

Not seq compact discrete

Closed & bounded.

Explanation

Closed & bounded.

9) Which sets seq compact in ℝ?

{1/n}∪{0}

(0,1)

[0,1]∪[5,6]

(−∞,0]

Closed & bounded.

Explanation

Closed & bounded.

Submit

10) Which ensure sequential compactness in metric space?

Totally bounded

Closed

Open

Bounded

Complete+totally bounded.

Explanation

Complete+totally bounded.

Submit

11) Which sets fail sequential compact?

(0,1)∪(2,3)

[0,1]{1/2}

[0,1]

{n}

Not closed or unbounded.

Explanation

Not closed or unbounded.

Submit

12) A set seq compact if:

All sequences converge

It is closed

It is bounded

Every sequence has conv subseq

Definition.

Explanation

Definition.

13) Which subsets have every seq with conv subseq staying inside?

[1,3]

[1,3)∪{3+1/n}

[−5,5]{0}

[−5,5]

Closed & bounded.

Explanation

Closed & bounded.

Submit

14) Which sets contain sequence lacking conv subseq inside?

(0,1)

{1,1/2,...}

[0,1]

{1/n}

Limit outside set.

Explanation

Limit outside set.

Submit

15) Sequentially compact in ℝ?

Closed intervals

Finite unions closed intervals

Infinite bounded closed sets

Any countable set

Closed & bounded.

Explanation

Closed & bounded.

Submit

Identifying Sequentially Compact Sets Quiz

1) A = {(-1)^n/n} sequentially compact because:

2)

What first name or nickname would you like us to use?

2) In ℝ, every sequentially compact set must contain all its limit points.

3) The set {1/n} is sequentially compact in ℝ.

4) Any infinite subset of ℕ is sequentially compact.

5) A finite subset of any metric space is always sequentially compact.

6) Which subset of ℝ is sequentially compact?

7) Which set is not sequentially compact?

8) (0,1] ∪ {2−1/n} is:

9) Which sets seq compact in ℝ?

10) Which ensure sequential compactness in metric space?

11) Which sets fail sequential compact?

12) A set seq compact if:

13) Which subsets have every seq with conv subseq staying inside?

14) Which sets contain sequence lacking conv subseq inside?

15) Sequentially compact in ℝ?