Convergence in Metric: Problem-Solving Quiz

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| Questions: 15 | Updated: Dec 15, 2025
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1) A sequence (xₙ) converges to x in a metric space if and only if:

Explanation

Convergence means distances from xₙ to x approach zero.

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About This Quiz
Convergence In Metric: Problem-solving Quiz - Quiz

Ready to put your convergence skills to the test? In this quiz, you’ll solve problems involving how sequences behave in different settings and how limit rules apply in metric spaces. You’ll practice identifying Cauchy sequences, working with subsequences, and analyzing how sequences interact through distance functions. Along the way, you’ll... see moreexplore examples where changing the metric changes the limit, where different subsequences approach different points, and where sequences converge only under special conditions. Each question helps sharpen your problem-solving skills and deepen your intuition for convergence in metric spaces.
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2) If a sequence converges in a metric space, then it must be:

Explanation

A convergent sequence must lie within some ball around the limit, making it bounded.

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3) Which of the following are necessary properties of a convergent sequence?

Explanation

Convergent sequences are always Cauchy, have unique limits, and are bounded.

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4) If xₙ → x, which must also converge?

Explanation

Distance to a fixed point varies continuously and must converge.

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5) Which describe basic properties of limits in metric spaces?

Explanation

Limits are unique and the definition depends on the metric.

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6) Which is always true if xₙ → x?

Explanation

This is the ε–N definition of convergence.

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7) In which metric spaces do all sequences converge only if they are eventually constant?

Explanation

These spaces force convergence only when the sequence eventually stabilizes at the limit.

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8) Let xₙ → x and y be fixed. Then:

Explanation

Distance to a fixed point converges because the metric is continuous.

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9) Which statements correctly identify how changing metrics affects convergence?

Explanation

Equivalent metrics preserve convergence; nonequivalent ones may alter it; identical topologies ensure identical convergence.

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10) Suppose xₙ → x. Which must be true?

Explanation

Every subsequence inherits the tail behavior toward x.

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11) A sequence has two subsequences converging to different limits. What can we conclude?

Explanation

Convergent sequences have unique limits; having two distinct limits contradicts convergence.

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12) Which properties are not guaranteed by convergence?

Explanation

Convergence does not require monotone, injective, or periodic behavior. Boundedness is guaranteed.

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13) Which statements are true regarding Cauchy and convergent sequences?

Explanation

Convergent → Cauchy; incomplete spaces may fail to converge; complete spaces guarantee convergence.

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14) A sequence (xₙ) satisfies: for every ε > 0, infinitely many n satisfy d(xₙ, x) < ε.

Explanation

Infinitely many terms approaching x does not imply eventual proximity; the sequence may oscillate.

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15) Which statements correctly describe convergent sequences?

Explanation

Convergent sequences can have infinitely many distinct values, and the subsequence property in C characterizes convergence.

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A sequence (xₙ) converges to x in a metric space if and only if:
If a sequence converges in a metric space, then it must be:
Which of the following are necessary properties of a convergent...
If xₙ → x, which must also converge?
Which describe basic properties of limits in metric spaces?
Which is always true if xₙ → x?
In which metric spaces do all sequences converge only if they are...
Let xₙ → x and y be fixed. Then:
Which statements correctly identify how changing metrics affects...
Suppose xₙ → x. Which must be true?
A sequence has two subsequences converging to different limits. What...
Which properties are not guaranteed by convergence?
Which statements are true regarding Cauchy and convergent sequences?
A sequence (xₙ) satisfies: for every ε > 0, infinitely many n...
Which statements correctly describe convergent sequences?
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