Convergence In Metric: Problem-Solving Quiz

1) A sequence (xₙ) converges to x in a metric space if and only if:

Xₙ = x for all n

D(xₙ, x) → 0

D(xₙ, x) is always less than 1

D(xₙ, x) is decreasing

Convergence means distances from xₙ to x approach zero.

Explanation

Convergence means distances from xₙ to x approach zero.

2) If a sequence converges in a metric space, then it must be:

Unbounded

Bounded

Eventually constant

Strictly increasing

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

Explanation

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

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

It is Cauchy

It has a unique limit

It is monotone

It is bounded

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

Explanation

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

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

D(xₙ, y) for any fixed y ∈ X

D(xₙ, yₙ) for any sequence yₙ

Xₙ₊₁

None of the above

Distance to a fixed point varies continuously and must converge.

Explanation

Distance to a fixed point varies continuously and must converge.

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

Limits are unique

Every subsequence has the same limit

Convergence depends on the metric

Convergence implies divergence of subsequences

Limits are unique and the definition depends on the metric.

Explanation

Limits are unique and the definition depends on the metric.

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6) Which is always true if xₙ → x?

D(xₙ, y) → 0 for some y ≠ x

D(xₙ, x) eventually becomes constant

D(xₙ, x) < ε for all sufficiently large n

(xₙ) is eventually periodic

This is the ε–N definition of convergence.

Explanation

This is the ε–N definition of convergence.

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

Discrete metric space

Euclidean space

Metric d(x,y)=min(1,|x−y|)

Any space where balls of radius < 1 contain only one point

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

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:

D(xₙ, y) diverges

D(xₙ, y) converges

D(xₙ, y) oscillates infinitely

Convergence cannot be determined

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

Explanation

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

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

Two equivalent metrics give the same convergent sequences

Nonequivalent metrics may give different convergence behaviors

If all open sets are the same, convergence is the same

Changing the metric always changes the limit

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

Any subsequence converges to x

Any subsequence diverges

Only one subsequence converges

No subsequence converges

Every subsequence inherits the tail behavior toward x.

Explanation

Every subsequence inherits the tail behavior toward x.

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

The sequence is convergent

The sequence has no limit

The sequence is bounded

The sequence is eventually constant

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

Explanation

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

12) Which properties are not guaranteed by convergence?

Monotonicity

Injectivity

Periodicity

Boundedness

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

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?

Every Cauchy sequence converges in any metric space

Every convergent sequence is Cauchy

Incomplete metric spaces may have non-convergent Cauchy sequences

Completeness guarantees all Cauchy sequences converge

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

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.

The sequence must converge to x

The sequence does not necessarily converge

The sequence must diverge

The sequence must be bounded

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

Explanation

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

15) Which statements correctly describe convergent sequences?

Every convergent sequence has a subsequence that is eventually constant

A convergent sequence may have infinitely many distinct values

A sequence converges to x iff every subsequence has a further subsequence converging to x

If a sequence has a convergent subsequence, then the entire sequence must converge

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

Explanation

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

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Convergence in Metric: Problem-Solving Quiz