Convergence In Metric: Concept Mastery Quiz

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1) If a sequence (xₙ) converges to x in a metric space (X,d), then d(xₙ, x) → 0.

True

False

True, because the definition of convergence in a metric space is exactly that d(xₙ, x) becomes arbitrarily small as n → ∞.

Explanation

True, because the definition of convergence in a metric space is exactly that d(xₙ, x) becomes arbitrarily small as n → ∞.

About This Quiz

Convergence In Metric: Concept Mastery Quiz - Quiz

Are you ready to explore how sequences behave in metric spaces? In this quiz, you’ll learn what it really means for a sequence to get closer and closer to a point using the distance function of the space. You’ll work with examples from ℝ, ℝ², discrete metrics, and even unusual... see moremetrics to see how convergence can look different depending on the distance used. By practicing formal definitions, comparing limits across metrics, and identifying which sequences converge (and why), you’ll build confidence in recognizing convergence in any metric space. Get ready to see how the simple idea “d(xₙ, x) → 0” becomes a powerful tool in understanding mathematical structure!
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2) A convergent sequence in any metric space must be bounded.

True

False

True, because all terms eventually lie within some ball around the limit, and finitely many initial terms can be bounded.

Explanation

True, because all terms eventually lie within some ball around the limit, and finitely many initial terms can be bounded.

3) Which best describes convergence of xₙ to x in a metric space (X,d)?

Xₙ = x for all sufficiently large n

D(xₙ, x) < ε for some fixed ε > 0

For every ε > 0, there exists N such that d(xₙ, x) < ε for all n ≥ N

Xₙ eventually becomes constant

This is the formal definition of convergence in a metric space.

Explanation

This is the formal definition of convergence in a metric space.

4) In (ℝ, d(x,y)=|x−y|), which sequence converges to 0?

Xₙ = n

Xₙ = 1/n

Xₙ = (−1)ⁿ

Xₙ = n²

1/n → 0, while the others diverge or oscillate.

Explanation

1/n → 0, while the others diverge or oscillate.

5) Consider X = ℝ² with the Euclidean metric. Which sequence converges to (0,0)?

Xₙ = (1, 1/n)

Xₙ = (1/n, 1/n)

Xₙ = (1/n, 1)

Xₙ = (n, 1/n)

Both coordinates go to 0, so the vector approaches (0,0).

Explanation

Both coordinates go to 0, so the vector approaches (0,0).

6) Let (X,d) be any metric space. If xₙ → x and xₙ → y, then:

X = y

Xₙ is constant

D(x,y) = 1

Convergence is not well-defined

Limits in metric spaces are unique; if a sequence converges to x and y, then x = y.

Explanation

Limits in metric spaces are unique; if a sequence converges to x and y, then x = y.

7) If d₁ and d₂ are equivalent metrics on the same set, then they generate the same convergent sequences.

True

False

True, because equivalent metrics induce the same topology, hence the same notion of convergence.

Explanation

True, because equivalent metrics induce the same topology, hence the same notion of convergence.

8) Let (xₙ) be a sequence with d(xₙ,x) → L > 0. Then the sequence:

Converges to x

Diverges

Converges to every point within distance L

Converges to a point y with d(x,y) = L

Convergence to x would require the distance to approach 0, not a positive number.

Explanation

Convergence to x would require the distance to approach 0, not a positive number.

9) In the metric d(x,y)=|x−y| on ℝ, which statement is true?

If xₙ → x, then |xₙ| → |x|

If |xₙ| → |x|, then xₙ → x

Convergence is preserved under negation: if xₙ → x, then −xₙ → −x

A sequence can have more than one limit

Absolute value is continuous, so |xₙ| → |x|, and negation preserves limits. Sequences in ℝ have unique limits.

Explanation

Absolute value is continuous, so |xₙ| → |x|, and negation preserves limits. Sequences in ℝ have unique limits.

10) In the metric space ℝ with d(x,y)=|x−y|^(1/2), a sequence (xₙ) converges to x iff:

|xₙ − x| (1/2) → 0

|xₙ − x| → 0

Both A and B

Neither A nor B

In a metric space defined by the distance function d(x,y) = |x - y|(1/2), a sequence converges to a limit x if the distance between the sequence terms and the limit approaches zero. This is expressed in two forms: the first is the distance raised to the power of (1/2) approaching zero, and the second is the distance itself approaching zero. Both conditions are equivalent, hence the correct option is 'Both A and B'.

Explanation

11) Let X = ℝ with the discrete metric d(x,y)=1 for x ≠ y. Which sequences converge?

Any bounded sequence

Only eventually constant sequences

Any Cauchy sequence

No sequences converge

A sequence converges only if eventually all terms equal the limit, since distances cannot shrink except by equality.

Explanation

A sequence converges only if eventually all terms equal the limit, since distances cannot shrink except by equality.

12) If xₙ is Cauchy in a metric space, then it must converge.

True

False

False, because only complete metric spaces guarantee that every Cauchy sequence converges.

Explanation

False, because only complete metric spaces guarantee that every Cauchy sequence converges.

13) Let X = ℝ with metric d(x,y)= $\frac {|x-y|} {1\, +\, |x-y|}$ . Which statement is true?

Convergence under this metric is different from the standard metric

A sequence converges in this metric iff it converges in the standard metric

No sequence converges in this metric

Only bounded sequences converge

The function t ↦ t/(1+t) is continuous and strictly increasing on [0,∞), so convergence is equivalent to standard convergence.

Explanation

The function t ↦ t/(1+t) is continuous and strictly increasing on [0,∞), so convergence is equivalent to standard convergence.

14) Let (xₙ) be a sequence such that for every ε>0, infinitely many n satisfy d(xₙ, x) < ε. Then the sequence:

Must converge to x

Does not necessarily converge

Must be eventually constant

Must diverge

Infinitely many terms close to x does not ensure that all sufficiently large terms stay close; the sequence may oscillate.

Explanation

Infinitely many terms close to x does not ensure that all sufficiently large terms stay close; the sequence may oscillate.

15) If (xₙ) → x and (yₙ) → y in a metric space (X,d), then d(xₙ, yₙ) → d(x, y).

True

False

True, because the metric function is continuous, so applying it to convergent pairs preserves limits.

Explanation

True, because the metric function is continuous, so applying it to convergent pairs preserves limits.

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Convergence in Metric: Concept Mastery Quiz