Exercise · ZW-BOLZANO-WEIERSTRASS-THEOREM-008 · ★★☆ Intermediate
If \((a_n)\) is bounded and all its convergent subsequences have limit \(4\), what follows?
Choose one option.
Show answer
- A. It converges to \(4\)
Explanation
If it did not converge to \(4\), there would be \(\varepsilon>0\) and infinitely many terms with \(|a_n-4|\ge\varepsilon\); by Bolzano-Weierstrass they would give a convergent subsequence with limit different from \(4\), a contradiction.
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-005 Which of these statements is true?
- 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-009 Why does the Bolzano-Weierstrass theorem fail if only rational numbers are allowed?
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