Boundedness: Concept Mastery Quiz

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Quizzes Created: 7682 | Total Attempts: 9,547,133
| Questions: 15 | Updated: Dec 15, 2025
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1) If a set A in a metric space (X,d) is bounded, then there exists M > 0 such that d(x,y) ≤ M for all x,y ∈ A.

Explanation

True, because boundedness means all pairwise distances in A are bounded by some constant.
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About This Quiz
Boundedness: Concept Mastery Quiz - Quiz

Are you ready to explore how metric spaces measure “size”? In this quiz, you’ll learn what it means for a set to be bounded — that everything inside it stays within some fixed distance. You’ll work with examples from ℝ, ℝ², and general metric spaces to see how sets behave... see moreunder different metrics. You’ll practice identifying bounded and unbounded sets, compare unions and intersections, and test how sequences relate to boundedness. By the end, you’ll understand how bounding works and why it’s an essential building block for compactness and continuity in metric spaces!
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2) Which of the following sets in ℝ with the usual metric is bounded?

Explanation

[0,5] lies within a finite interval; the others extend infinitely.

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3) Every finite subset of a metric space is bounded.

Explanation

True, because finitely many distances achieve a maximum value.

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4) Let A = {(x,y) ∈ ℝ² : x² + y² ≤ 4}. Which statement is true?

Explanation

It is a closed disk of radius 2, so it is bounded and closed.

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5) If a set A in a metric space is bounded, then every subset of A is also bounded.

Explanation

True, because all distances in a subset are distances from A, which are already bounded.

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6) Let (X,d) be a metric space and let A, B ⊂ X be bounded. Which must be true?

Explanation

The intersection is contained in each set, so it inherits boundedness. The union may not be bounded.

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7) A set A is bounded in a metric space (X,d) if there exists x₀ ∈ X and r > 0 such that A ⊂ B(x₀,r).

Explanation

True, because this is an equivalent definition of boundedness.

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8) Which set is NOT necessarily bounded?

Explanation

A bounded × bounded set in ℝ may not be bounded under some metrics.

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9) If a sequence (xₙ) in a metric space is bounded, then the set {xₙ : n ∈ ℕ} is bounded.

Explanation

True, because boundedness of the sequence means all its elements lie in some bounded ball.

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10) Let A = { xₙ = n/(n+1) : n ∈ ℕ } in ℝ. Which is true?

Explanation

All values lie between 0 and 1.

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11) A bounded set in a metric space is always compact.

Explanation

False, compactness also requires closedness and completeness.

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12) Which condition implies A ⊂ X is bounded?

Explanation

Any of these conditions ensures boundedness.

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13) If a set A ⊂ ℝ is bounded and its supremum exists, then sup A ∈ A.

Explanation

False, sup A need not belong to A.

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14) Let Aₙ = { x ∈ ℝ : |x| < n } for  n∈N . Which statement is true?

Explanation

Each Aₙ is bounded, but their union is ℝ, unbounded.

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15) The empty set is considered bounded in any metric space.

Explanation

True, because boundedness requires ∃M such that all distances in A are ≤ M, which is vacuously true for the empty set.

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If a set A in a metric space (X,d) is bounded, then there exists M...
Which of the following sets in ℝ with the usual metric is bounded?
Every finite subset of a metric space is bounded.
Let A = {(x,y) ∈ ℝ² : x² + y² ≤ 4}. Which statement is true?
If a set A in a metric space is bounded, then every subset of A is...
Let (X,d) be a metric space and let A, B ⊂ X be bounded. Which must...
A set A is bounded in a metric space (X,d) if there exists x₀ ∈ X...
Which set is NOT necessarily bounded?
If a sequence (xₙ) in a metric space is bounded, then the set {xₙ...
Let A = { xₙ = n/(n+1) : n ∈ ℕ } in ℝ. Which is true?
A bounded set in a metric space is always compact.
Which condition implies A ⊂ X is bounded?
If a set A ⊂ ℝ is bounded and its supremum exists, then sup A ∈...
Let Aₙ = { x ∈ ℝ : |x| < n } for  n∈N . Which...
The empty set is considered bounded in any metric space.
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