The Cauchy criterion is especially powerful for proving convergence in cases where:
AThe sequence is monotone and bounded, so the limit equals the supremum of the sequence
BThe sequence oscillates but eventually enters a bounded region
CA limit exists but cannot be written in closed form — the Cauchy criterion requires no knowledge of the limit beforehand
DThe sequence is defined recursively and each term can be computed explicitly
The standard convergence definition requires knowing a candidate limit L and showing terms eventually stay within ε of L. Many deep existence proofs in analysis deal with limits that cannot be expressed in closed form — the whole point is to prove the limit exists without constructing it. The Cauchy criterion sidesteps this by asking only whether later terms cluster together, with no reference to any external target. Options A and D describe situations where the limit can be found or computed, which don't require the Cauchy criterion's special advantage.
Question 2 Multiple Choice
The sequence 3, 3.1, 3.14, 3.141, 3.1415, … (successive rational approximations of π) is:
ACauchy in ℚ and convergent in ℚ, since the terms are clearly approaching a limit
BCauchy in ℚ but not convergent in ℚ, since π is irrational and does not exist in ℚ
CNot Cauchy in ℚ because the differences between consecutive terms never reach exactly zero
DConvergent in ℚ by the completeness of the rational numbers
The sequence is Cauchy in ℚ: for any ε > 0, all terms beyond some index are within ε of each other (they share many decimal places). But the only candidate limit is π, which is irrational — it is not in ℚ. A Cauchy sequence in ℚ has no obligation to converge in ℚ. This is the essential incompleteness of ℚ: it has 'holes' at the irrationals. Option D is wrong — ℚ is NOT complete, which is exactly the point of this example.
Question 3 True / False
A sequence can be Cauchy without knowing its limit — the Cauchy property depends only on the mutual distances between terms, with no reference to any external target value.
TTrue
FFalse
Answer: True
This is the defining feature that makes the Cauchy criterion useful. The condition |aₙ - aₘ| < ε for all n, m > N is an intrinsic property of the sequence: it only looks at how terms relate to each other, not to any external point. By contrast, the standard convergence definition |aₙ - L| < ε requires specifying L in advance. In ℝ, the two conditions are equivalent — but the Cauchy formulation is often the one you can verify when the limit is unknown.
Question 4 True / False
Completeness is a theorem that can be derived from axioms shared by both ℝ and ℚ — it is not a fundamental property that distinguishes one number system from another.
TTrue
FFalse
Answer: False
Completeness is one of the *defining* properties of ℝ, not a consequence of more basic axioms. ℚ satisfies the ordered field axioms just as ℝ does, but ℚ is not complete. Completeness is the additional axiom that distinguishes ℝ from ℚ — it was included in the construction of ℝ (via Dedekind cuts or Cauchy completion) precisely to plug the holes that ℚ leaves. You cannot prove completeness of ℝ from axioms that ℚ also satisfies.
Question 5 Short Answer
What does it mean for a space to be 'complete,' and why does the rational number system ℚ fail this property?
Think about your answer, then reveal below.
Model answer: A space is complete if every Cauchy sequence in it converges to a point in it — there are no 'holes' that Cauchy sequences can fall into. ℚ fails completeness because it contains Cauchy sequences whose natural limit is irrational. For example, the decimal approximations of √2 form a sequence where terms get arbitrarily close to each other (Cauchy), but their limit √2 is not rational — it does not exist in ℚ. The sequence 'wants' to converge but has nowhere to go within ℚ. ℝ was constructed specifically to ensure every such hole is filled.
Completeness is not an abstract nicety — it is what makes analysis work. Without completeness, you cannot be sure that objects defined as limits actually exist. The entire architecture of real analysis (intermediate value theorem, Riemann integral, etc.) depends on ℝ having no holes.