The sequence 1, 1.4, 1.41, 1.414, 1.4142, … (rational approximations to √2) is considered in the metric space (ℚ, d) where d is the usual absolute value. Which statement is correct?
AIt converges in ℚ because the terms get arbitrarily close to each other
BIt is Cauchy in ℚ but does not converge in ℚ, because √2 is irrational and not in ℚ
CIt is not Cauchy in ℚ because it has no limit in ℚ
DIt converges in ℚ to the rational number closest to √2
The sequence is Cauchy because its terms get arbitrarily close to one another — the Cauchy property is intrinsic to the sequence and does not require a limit. But √2 ∉ ℚ, so the sequence has no limit in ℚ. This is the canonical example showing that Cauchy ≠ convergent in incomplete spaces. Option C is the key misconception: the absence of a limit does not make the sequence non-Cauchy.
Question 2 Multiple Choice
In which of the following spaces does every Cauchy sequence converge?
Aℚ with the usual metric
BThe open interval (0, 1) with the usual metric
Cℝ with the usual metric
DAny metric space that contains a dense subset of ℝ
ℝ is complete: every Cauchy sequence of real numbers converges to a real number. ℚ is incomplete (the √2 example). The open interval (0, 1) is incomplete because the sequence 1/2, 1/3, 1/4, … is Cauchy in (0, 1) but converges to 0, which is not in the space. Option D is false — density doesn't guarantee completeness.
Question 3 True / False
Every Cauchy sequence in a metric space is bounded.
TTrue
FFalse
Answer: True
If (xₙ) is Cauchy, then for ε = 1 there exists N such that d(xₙ, xₘ) < 1 for all n, m > N. In particular, all terms beyond index N lie within distance 1 of x_{N+1}. The finitely many terms up to index N are also at finite distance from x_{N+1}. Taking the maximum of these finitely many distances plus 1 gives a global bound. So boundedness is a consequence of the Cauchy property, not an independent assumption.
Question 4 True / False
If a sequence converges in a metric space, the space should be complete.
TTrue
FFalse
Answer: False
Convergence of some sequences does not imply completeness. A space is complete only if every Cauchy sequence converges. ℚ contains many convergent sequences (e.g., 1, 1, 1, … converges to 1 ∈ ℚ), yet ℚ is not complete because the sequence of rational approximations to √2 is Cauchy but does not converge in ℚ. Completeness is a global property of the space, not a statement about individual sequences.
Question 5 Short Answer
What does it mean for a metric space to be 'complete,' and why can the same sequence be Cauchy in one space but fail to converge in another?
Think about your answer, then reveal below.
Model answer: A metric space is complete if every Cauchy sequence in it converges to a point within the space. The same sequence can be Cauchy in multiple spaces (since the Cauchy property depends only on mutual distances between terms), but convergence requires the limit point to exist in the space. If the limit point is 'missing' — as √2 is missing from ℚ — the sequence is Cauchy but non-convergent, revealing a 'hole' in the space.
This is the heart of the topic: Cauchy sequences detect clustering behavior without naming a limit, while convergence requires the limit to actually exist in the space. The same sequence of rational approximations to √2 is Cauchy in both ℚ and ℝ, but it only converges in ℝ (where √2 lives). Completeness is the property that rules out such holes, and it is the key hypothesis in many major theorems of analysis.