Boundedness: Problem-Solving Quiz

1) Let A = {1/n : n ∈ ℝ} in ℝ. Which statement is correct?

A. A is unbounded

B. A is bounded above but not below

C. A is bounded below but not above

D. A is bounded both above and below

Values lie between 0 and 1, so A is bounded on both sides.

Explanation

Values lie between 0 and 1, so A is bounded on both sides.

2) Let A = { (−1)ⁿ + 1/n : n ∈ ℕ }. Which statements are true?

A. A is bounded

B. sup A = 1 + 1/1 = 2

C. sup A = −1 − 1/2

D. A oscillates but stays within a finite interval

The sequence oscillates near ±1 but stays in a finite interval, so it is bounded.

Explanation

The sequence oscillates near ±1 but stays in a finite interval, so it is bounded.

3) A metric space (X,d) contains sets A and B. If A is bounded and B is finite, which is true?

A. A ∪ B is bounded

B. A ∩ B is unbounded

C. A − B is unbounded

D. None of the above

A finite set is bounded, and the union of two bounded sets is also bounded.

Explanation

A finite set is bounded, and the union of two bounded sets is also bounded.

4) Let A = { n/(n+1) : n ∈ ℕ }. Which statements are true?

A. A is bounded

B. sup A = 1

C. inf A = 0

D. A contains 1

n/(n+1) approaches 1 from below and stays above 0, so it is bounded; 1 is not included.

Explanation

n/(n+1) approaches 1 from below and stays above 0, so it is bounded; 1 is not included.

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5) Let Aₙ = { x ∈ ℝ : |x| < n }, and consider ⋃ Aₙ. What can we say?

A. Bounded

B. Unbounded

C. Finite

D. Empty

The union over all n includes all real numbers, so it is unbounded.

Explanation

The union over all n includes all real numbers, so it is unbounded.

6) Let A, B ⊂ X be bounded sets. Which of the following are always bounded?

A. A ∩ B

B. A ∪ B

C. A + B = {x + y : x ∈ A, y ∈ B}

D. A \ B

Intersection, union, and Minkowski sum of bounded sets remain bounded. A \ B is not guaranteed bounded.

Explanation

Intersection, union, and Minkowski sum of bounded sets remain bounded. A \ B is not guaranteed bounded.

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7) If a bounded set A in (X,d) contains infinitely many points, which is true?

A. A must be finite

B. A is bounded but not necessarily closed

C. A is compact

D. A is unbounded

Bounded infinite sets may fail to be closed and are not necessarily compact.

Explanation

Bounded infinite sets may fail to be closed and are not necessarily compact.

8) Let A ⊂ ℝ² be a bounded set. Which statements are always true?

A. The diameter of A is finite

B. Any subset of A is bounded

C. The closure of A is bounded

D. The interior of A may be unbounded

Bounded sets have finite diameter; subsets and closures remain bounded. Interior cannot exceed the set.

Explanation

Bounded sets have finite diameter; subsets and closures remain bounded. Interior cannot exceed the set.

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9) A company tracks distances of trucks from a warehouse: 0 km ≤ distance ≤ 150 km. What can be said?

A. The set of distances is bounded

B. The set is unbounded

C. Only bounded above

D. Only bounded below

Distances lie in a finite interval [0,150].

Explanation

Distances lie in a finite interval [0,150].

10) Let A = { x ∈ ℝ : |x − 2| < 5 } and B = { x ∈ ℝ : |x + 3| ≤ 4 }. What is true about A ∪ B?

A. Unbounded

B. Bounded

C. Empty

D. Singleton

A = (−3,7), B = [−7,1]. Union lies within [−7,7], bounded.

Explanation

A = (−3,7), B = [−7,1]. Union lies within [−7,7], bounded.

11) A bounded set A ⊂ ℝ has sup A = 5 and inf A = −2. What is the maximum possible distance between any two points in A?

A. 3

B. 5

C. 7

D. 10

The max distance is 5 - (-2) = 7.

Explanation

The max distance is 5 - (-2) = 7.

12) Let A = { 1/n : n ∈ ℕ }. Which statements are correct?

A. A is bounded

B. A has 0 as a limit point

C. A contains its supremum

D. A is finite

Values are between 0 and 1, with 0 a limit point. Does not include 1, infinite set.

Explanation

Values are between 0 and 1, with 0 a limit point. Does not include 1, infinite set.

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13) In a metric space (X,d), if A is bounded and f : X → ℝ is Lipschitz with constant L > 0, then f(A) is:

A. Bounded

B. Unbounded

C. Empty

D. Bounded only if A is closed

Lipschitz maps scale distances, preserving boundedness.

Explanation

Lipschitz maps scale distances, preserving boundedness.

14) Let A = {(x,y) ∈ ℝ² : x² + y² < 16} and B = {(x,y) ∈ ℝ² : (x−3)² + (y+2)² < 9}. What is true about A ∪ B?

A. Unbounded

B. Bounded

C. Empty

D. Contains all of ℝ²

Both are finite-radius disks; union remains bounded.

Explanation

Both are finite-radius disks; union remains bounded.