The sequence a_n = sin(n) is bounded (|sin(n)| ≤ 1 for all n) but does not converge. What does the Bolzano-Weierstrass theorem guarantee about this sequence?
AThe sequence must eventually converge since it is bounded
BThe sequence must become monotone at some point, after which it converges
CThe sequence has a convergent subsequence, even though the full sequence diverges
DThe sequence has at most one accumulation point in [−1, 1]
Bolzano-Weierstrass guarantees that every bounded sequence has a *convergent subsequence*, not that the sequence itself converges. sin(n) is bounded and nowhere near convergent — it oscillates without settling — but there exist infinite subsequences of its values that do converge. This distinction between 'the sequence converges' and 'a subsequence converges' is the essential content of the theorem.
Question 2 Multiple Choice
The proof of Bolzano-Weierstrass by repeated interval bisection uses which fundamental property of ℝ at its crucial step?
AThe archimedean property — for any real number, there exists a larger integer
BThe density of ℚ — between any two reals there is a rational number
CCompleteness — the nested interval property guarantees that the intersection of nested closed intervals contains a point of ℝ
DThe uncountability of ℝ — the sequence cannot exhaust all real numbers
The bisection produces nested closed intervals [a_n, b_n] with lengths shrinking to zero. The nested interval property — a consequence of completeness — guarantees their intersection contains exactly one real point L. This is where completeness is used: in ℚ, the same bisection might produce intervals whose intersection is irrational, so no rational limit point exists. Bolzano-Weierstrass is a theorem about ℝ precisely because ℝ is complete.
Question 3 True / False
The Bolzano-Weierstrass theorem holds in the rational numbers ℚ: most bounded sequence of rationals has a convergent subsequence converging to a rational limit.
TTrue
FFalse
Answer: False
This fails because ℚ is not complete — it has 'holes.' A sequence of rationals can converge to an irrational number (e.g., the decimal approximations 1, 1.4, 1.41, 1.414, ... converge to √2). Any subsequence of this rational sequence also converges to √2, which is not in ℚ. The theorem holds in ℝ because ℝ has no such holes — every Cauchy sequence (and hence every limit point produced by the bisection argument) corresponds to an actual real number.
Question 4 True / False
Bolzano-Weierstrass implies that most bounded sequence in ℝ converges.
TTrue
FFalse
Answer: False
A bounded sequence need not converge — it need only have a convergent subsequence. sin(n) is a standard counterexample: bounded but not convergent. The theorem guarantees accumulation points, not convergence of the whole sequence. A bounded sequence converges if and only if it has exactly one accumulation point (equivalently, all its convergent subsequences share the same limit). Bolzano-Weierstrass guarantees at least one accumulation point; convergence requires at most one.
Question 5 Short Answer
Why does the Bolzano-Weierstrass theorem fail in ℚ but hold in ℝ, and what does this reveal about the role of completeness in the theorem?
Think about your answer, then reveal below.
Model answer: The bisection proof produces a limit point L as the intersection of nested intervals. In ℝ, completeness guarantees L exists as a real number, so every subsequence selected from those intervals converges to L within ℝ. In ℚ, L might be irrational — a 'hole' that ℚ does not contain — and the constructed subsequence converges to something outside the space. Bolzano-Weierstrass is therefore not a theorem about sequences per se; it is a theorem about the completeness of ℝ expressed in sequential language. The theorem fails in any incomplete metric space where limit points can fall outside the space.
This connection explains why Bolzano-Weierstrass is equivalent to the Heine-Borel theorem in ℝ: both are ways of asserting that closed bounded subsets of ℝ are 'compact' — they retain all their limit points. Completeness is the structural property that makes this possible.