Completeness: Problem-Solving Quiz

1) If a sequence in a metric space is Cauchy and the space is complete, then the sequence converges to a unique limit in that space.

True

False

True, because completeness ensures every Cauchy sequence converges, and limits in metric spaces are always unique.

Explanation

True, because completeness ensures every Cauchy sequence converges, and limits in metric spaces are always unique.

2) Let X = {0,1,2,3} with the usual distance. Which statement is correct?

X is incomplete

X is complete

X contains a Cauchy sequence that diverges

X has a Cauchy sequence that converges outside X

Finite metric spaces are always complete, since any Cauchy sequence must eventually repeat a value and thus converge.

Explanation

Finite metric spaces are always complete, since any Cauchy sequence must eventually repeat a value and thus converge.

3) If A is an incomplete subset of a complete metric space X, then there exists a Cauchy sequence in A that converges to a point in X \ A.

True

False

True, because incompleteness means some Cauchy sequence in A fails to converge in A, but must converge in the larger complete space X to a point outside A.

Explanation

True, because incompleteness means some Cauchy sequence in A fails to converge in A, but must converge in the larger complete space X to a point outside A.

4) Let (0,1] with the usual metric. A student claims every Cauchy sequence in this set converges in the set. Is this correct?

Yes, because all sequences are bounded

No, because sequences can converge to 0, which is outside the set

Yes, because it is an interval

No, because the set is finite

A Cauchy sequence like 1/n lies in (0,1] but converges to 0, which is not in the set.

Explanation

A Cauchy sequence like 1/n lies in (0,1] but converges to 0, which is not in the set.

5) All bounded sequences in a complete metric space converge.

True

False

False, boundedness alone does not imply convergence (e.g., oscillating sequences).

Explanation

False, boundedness alone does not imply convergence (e.g., oscillating sequences).

6) Which reasoning strategy helps prove a subset A of a complete metric space X is complete?

Show A is closed in X

Show every sequence in A is bounded

Show A is open in X

Show A contains a finite number of points

A closed subset of a complete space is complete.

Explanation

A closed subset of a complete space is complete.

7) In ℝ² with the Euclidean metric, the open disk {(x,y): x² + y² < 1} is incomplete.

True

False

True, because sequences may converge to points on the boundary, which are not included.

Explanation

True, because sequences may converge to points on the boundary, which are not included.

8) Consider C[0,1] with the sup metric. Which reasoning shows that it is complete?

Every bounded sequence of functions converges

Every sequence has a pointwise limit

Every Cauchy sequence of functions converges uniformly to a continuous function

Continuous functions are bounded

Uniform limits of Cauchy sequences of continuous functions remain continuous.

Explanation

Uniform limits of Cauchy sequences of continuous functions remain continuous.

9) A contraction mapping on a complete metric space always has a unique fixed point.

True

False

True, by the Banach Fixed-Point Theorem.

Explanation

True, by the Banach Fixed-Point Theorem.

10) Let X = ℝ \ {0}. A Cauchy sequence x_n = 1/n is in X. What can we conclude?

The sequence converges in X

The sequence is not Cauchy

The sequence shows X is incomplete

The sequence is bounded but diverges

It converges to 0, which is not in X.

Explanation

It converges to 0, which is not in X.

11) Completeness is preserved under isometric embeddings.

True

False

False, because an isometric embedding need not be onto.

Explanation

False, because an isometric embedding need not be onto.

12) A student wants to prove ℚ is incomplete. Which reasoning is correct?

Every sequence in ℚ is bounded

There exists a Cauchy sequence in ℚ converging to an irrational number

ℚ is dense in ℝ

ℚ is countable

A rational sequence converging to √2 proves ℚ is incomplete.

Explanation

A rational sequence converging to √2 proves ℚ is incomplete.

13) If X is complete and f:X→X is Lipschitz with constant L<1, then iteration converges to a unique fixed point.

True

False

This is the contraction mapping theorem.

Explanation

This is the contraction mapping theorem.

14) Which subset of ℝ² is reasoning-wise complete?

Open unit square (0,1)x(0,1)

Closed unit square [0,1]x[0,1]

Open disk

Half-plane y>0

Closed subsets of ℝ² are complete.

Explanation

Closed subsets of ℝ² are complete.

15) A Cauchy sequence in an incomplete metric space may converge in a larger complete space containing it.

True

False

True, incompleteness means missing limit points.

Explanation

True, incompleteness means missing limit points.