Convergence In Metric: Property Identification Quiz

1) If a sequence xₙ → x in a metric space, then every subsequence also converges to x.

True

False

True, because subsequences inherit the tail behavior of the original sequence, so they must approach the same limit.

Explanation

True, because subsequences inherit the tail behavior of the original sequence, so they must approach the same limit.

2) Convergence of a sequence depends on the metric used.

True

False

True, because different metrics may define different distances, altering whether terms get arbitrarily close.

Explanation

True, because different metrics may define different distances, altering whether terms get arbitrarily close.

3) If xₙ → x in a metric space, which property always holds?

Xₙ becomes constant eventually

D(xₙ, x) becomes arbitrarily small

Xₙ → y for every y ∈ X

Xₙ must be monotonic

Convergence requires that distances from x shrink arbitrarily small.

Explanation

Convergence requires that distances from x shrink arbitrarily small.

4) In any metric space, a convergent sequence must be:

Bounded

Cauchy

Both A and B

Neither

Convergent sequences are always both bounded and Cauchy in any metric space.

Explanation

Convergent sequences are always both bounded and Cauchy in any metric space.

5) If xₙ → x, then d(xₙ, y) → d(x, y) for any fixed y ∈ X.

True

False

True, because the metric is continuous, so distance to a fixed point varies continuously with xₙ.

Explanation

True, because the metric is continuous, so distance to a fixed point varies continuously with xₙ.

6) Which of the following is always a property of convergent sequences in metric spaces?

Two convergent sequences must converge to the same limit

Convergence ensures the sequence is injective

A convergent sequence has a unique limit

A convergent sequence must be finite

Limits in metric spaces are unique.

Explanation

Limits in metric spaces are unique.

7) If a sequence is eventually inside every open ball centered at x, then it converges to x.

True

False

True, because this is equivalent to the ε–N definition of convergence.

Explanation

True, because this is equivalent to the ε–N definition of convergence.

8) If xₙ → x and the metric d is replaced by an equivalent metric d′, then:

The sequence may converge to a different limit

The sequence converges to the same limit under d′

The sequence diverges under d′

Convergence cannot be determined

Equivalent metrics generate the same convergent sequences and the same limits.

Explanation

Equivalent metrics generate the same convergent sequences and the same limits.

9) If two sequences converge to the same limit x, then the sequence formed by interleaving their terms also converges to x.

True

False

True, because each subsequence converges to x, so the combined sequence must also converge to x.

Explanation

True, because each subsequence converges to x, so the combined sequence must also converge to x.

10) In the discrete metric d(x,y)=1 for x≠y, which property of convergence is true?

Every bounded sequence converges

Only eventually constant sequences converge

A sequence converges iff it is Cauchy

No sequence converges

Convergence requires xₙ = x eventually, since the distance cannot shrink unless the terms equal the limit.

Explanation

Convergence requires xₙ = x eventually, since the distance cannot shrink unless the terms equal the limit.

11) Let xₙ → x and let yₙ be any bounded sequence. Which is true about d(xₙ, yₙ)?

It must converge

It must diverge

It may or may not converge

It must converge to zero

The distance may oscillate depending on yₙ; boundedness does not guarantee convergence.

Explanation

The distance may oscillate depending on yₙ; boundedness does not guarantee convergence.

12) A sequence with more than one convergent subsequence must converge.

True

False

False, because those subsequences could converge to different limits, which implies the original sequence is not convergent.

Explanation

False, because those subsequences could converge to different limits, which implies the original sequence is not convergent.

13) If xₙ → x and yₙ → y, which must hold?

D(xₙ, yₙ) stays constant for large n

D(xₙ, yₙ) → d(x, y)

Xₙ = yₙ eventually

D(xₙ, yₙ) → 0

Distances of convergent sequences converge to the distance of their limits by continuity of the metric.

Explanation

Distances of convergent sequences converge to the distance of their limits by continuity of the metric.

14) Every convergent sequence in a metric space has a subsequence that is eventually constant.

True

False

False, a convergent sequence may have infinitely many distinct values (e.g., 1/n).

Explanation

False, a convergent sequence may have infinitely many distinct values (e.g., 1/n).

15) Which statement characterizes a property not guaranteed by convergence?

The sequence is bounded

The sequence is Cauchy

The sequence is eventually in every neighborhood of the limit

The sequence is monotone

A convergent sequence need not be monotone.

Explanation

A convergent sequence need not be monotone.

Convergence in Metric: Property Identification Quiz