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

Every Cauchy sequence of functions converges uniformly to a continuous function

Every bounded sequence of functions converges

Every sequence has a pointwise limit

Continuous functions are bounded

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

The sequence shows X is incomplete

The sequence is bounded but diverges

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

Show every sequence in A is bounded

Show A contains a finite number of points

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

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

Every sequence in ℚ is bounded

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

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

Yes, because all sequences are bounded

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

X contains a Cauchy sequence that diverges

X has a Cauchy sequence that converges outside X

Completeness: Problem-Solving Quiz

Reviewed by Jede Crisle Cortes Davila

Jede Crisle Cortes Davila, Bachelor of Engineering |

Math Expert

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Jede Crisle D. is a mathematics subject matter expert specializing in Algebra, Geometry, and Calculus. She focuses on developing clear, solution-driven mathematical explanations and has strong experience with LaTeX-based math content. She holds a Bachelor’s degree in Electronics and Communications Engineering.

, Bachelor of Engineering

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