Exercise · ZW-BOLZANO-WEIERSTRASS-THEOREM-005 · ★★☆ Intermediate
Which of these statements is true?
Choose one option.
Show answer
- B. Every bounded sequence has a convergent subsequence
Explanation
That is the theorem. Counterexamples to the others: \((-1)^n\) is bounded and does not converge; \((-1)^n/n\) converges without being monotone; and \(1,2,1,3,1,4,\dots\) has the constant subsequence \(1\) and is unbounded.
Review the concept: Bolzano-Weierstrass theorem
Every bounded real sequence \((a_n)\) has at least one subsequence \((a_{n_k})\) that converges to a number in the set \(\mathbb R\) of reals.
\[\exists\,(n_k):\ \lim_{k\to\infty}a_{n_k}\in\mathbb R\]
More practice problems: bolzano-weierstrass theorem
- ZW-BOLZANO-WEIERSTRASS-THEOREM-001 Which of these sequences satisfies the hypothesis of the Bolzano-Weierstrass theorem?
- ZW-BOLZANO-WEIERSTRASS-THEOREM-002 Let a_n=(-1)^n. Which of these is a convergent subsequence?
- ZW-BOLZANO-WEIERSTRASS-THEOREM-003 For a_n=n, why does the Bolzano-Weierstrass theorem not guarantee a convergent subsequence?
- ZW-BOLZANO-WEIERSTRASS-THEOREM-004 Let a_n=(3n+1)/(n+1). To what number does every subsequence of (a_n) converge?
- ZW-BOLZANO-WEIERSTRASS-THEOREM-006 Let a_n=sin ((nπ)/(3)). How many distinct numbers are the limit of some subsequence?
- ZW-BOLZANO-WEIERSTRASS-THEOREM-007 In the bisection proof we start with [-8,8] and at each step keep the half with infinitely many ter…
- ZW-BOLZANO-WEIERSTRASS-THEOREM-008 If (a_n) is bounded and all its convergent subsequences have limit 4, what follows?
- ZW-BOLZANO-WEIERSTRASS-THEOREM-009 Why does the Bolzano-Weierstrass theorem fail if only rational numbers are allowed?
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