Boundedness: Basic Properties Quiz

1) Let A = { (−1)ⁿ+ 1/n : n ∈ ℕ }. Which is correct?

A is unbounded

A bounded above but not below

A bounded below but not above

A bounded both above and below

Terms oscillate near ±1, so all lie in a finite interval.

Explanation

Terms oscillate near ±1, so all lie in a finite interval.

2) A subset of a bounded set is always bounded.

True

False

True, because a subset cannot exceed the bounding ball of the original set.

Explanation

True, because a subset cannot exceed the bounding ball of the original set.

3) Let A ⊂ ℝ² be defined by A = {(x,y): |x| ≤ 2, |y| ≤ 3}. Which statement is true?

A unbounded

A bounded

A bounded only in x

A bounded only in y

It is contained in a finite rectangle.

Explanation

It is contained in a finite rectangle.

4) If A and B are bounded sets in a metric space, then A − B = {x − y : x ∈ A, y ∈ B} is also bounded.

True

False

True, because the distance between any x ∈ A and any y ∈ B is bounded by the sum of their bounds.

Explanation

True, because the distance between any x ∈ A and any y ∈ B is bounded by the sum of their bounds.

5) If a set A in a metric space (X,d) is bounded, then there exists some point x₀ ∈ X such that A ⊆ B(x₀, R) for some R > 0.

True

False

True, because this is an equivalent definition of boundedness: the set fits inside one finite-radius ball.

Explanation

True, because this is an equivalent definition of boundedness: the set fits inside one finite-radius ball.

6) Which of the following sets in ℝ with the usual metric is bounded?

(−∞, 1]

[2,7]

(−∞, 1)

(2,7)

Both [2,7] and (2,7) lie inside finite intervals; the others extend indefinitely.

Explanation

Both [2,7] and (2,7) lie inside finite intervals; the others extend indefinitely.

7) The union of two bounded sets in a metric space is always bounded.

True

False

True, because both sets lie in finite balls, and their union lies in a ball containing both radii.

Explanation

True, because both sets lie in finite balls, and their union lies in a ball containing both radii.

8) Let A = {x ∈ ℝ : |x − 3| < 5}. Which is correct?

A is bounded

A is unbounded

A is empty

A contains only integer points

A = (−2, 8), which fits in a finite interval, so it is bounded.

Explanation

A = (−2, 8), which fits in a finite interval, so it is bounded.

9) If a set is bounded, then all sequences contained in the set are also bounded.

True

False

True, because the entire set lies within a fixed radius, so every sequence in it must as well.

Explanation

True, because the entire set lies within a fixed radius, so every sequence in it must as well.

10) Let A = {(x,y) ∈ ℝ² : x² + y² < 9} and B = {(x,y) ∈ ℝ² : x² + y² ≥ 4}. Which is true?

A is bounded, B is bounded

A is bounded, B is unbounded

A is unbounded, B is bounded

Both A and B are unbounded

A is a disk of radius 3 (bounded). B extends outward infinitely (unbounded).

Explanation

A is a disk of radius 3 (bounded). B extends outward infinitely (unbounded).

11) If a set A in a metric space is bounded, then its closure cl(A) is also bounded.

True

False

True, because the closure adds only limit points, which must lie within the same bounded region.

Explanation

True, because the closure adds only limit points, which must lie within the same bounded region.

12) Let A, B ⊂ X be bounded sets. Which of the following is NOT necessarily bounded?

A ∩ B

A ∪ B

A×B

None of the above

In some metrics, a Cartesian product of bounded sets may not be bounded.

Explanation

In some metrics, a Cartesian product of bounded sets may not be bounded.

13) The empty set is bounded in any metric space.

True

False

True, vacuously, since there are no distances to exceed a bound.

Explanation

True, vacuously, since there are no distances to exceed a bound.

14) Which statement is true about finite sets in metric spaces?

All finite sets are unbounded

All finite sets are bounded

Some finite sets are unbounded

None

Finite sets have only finitely many distances, so always bounded.

Explanation

Finite sets have only finitely many distances, so always bounded.

15) A bounded set in ℝ can have an infinite number of elements.

True

False

True, intervals like (0,1) are bounded and infinite.

Explanation

True, intervals like (0,1) are bounded and infinite.