Exercise · ZW-SEQUENTIAL-COMPACTNESS-METRIC-SPACES-007 · ★★☆ Intermediate
Let \(x_n=(-1)^n\,\dfrac{n}{n+1}\) in \([-1,1]\). To which values do its convergent subsequences converge?
Choose one option.
Show answer
- D. To \(1\) and to \(-1\)
Explanation
The even terms are \(\tfrac{n}{n+1}\to1\) and the odd ones \(-\tfrac{n}{n+1}\to-1\). Every convergent subsequence has infinitely many terms of one kind, so its limit is \(1\) or \(-1\); the sequence itself does not converge.
Review the concept: Sequential compactness
A metric space \(X\) is sequentially compact if every sequence \((x_n)\) in \(X\) has a subsequence \((x_{n_k})\) converging to a point \(x\in X\).
\[\forall (x_n)\subseteq X\ \ \exists\, n_1
More practice problems: sequential compactness
- ZW-SEQUENTIAL-COMPACTNESS-METRIC-SPACES-001 Which of these spaces, with the usual distance, is sequentially compact?
- ZW-SEQUENTIAL-COMPACTNESS-METRIC-SPACES-002 Why is ℝ not sequentially compact?
- ZW-SEQUENTIAL-COMPACTNESS-METRIC-SPACES-003 To prove that (0,1) is not sequentially compact, which sequence works?
- ZW-SEQUENTIAL-COMPACTNESS-METRIC-SPACES-004 In X=(1,2] with the usual distance, does the sequence x_n=1+tfrac1n have a subsequence converging i…
- ZW-SEQUENTIAL-COMPACTNESS-METRIC-SPACES-005 Which of these subsets of ℝ^2, with the Euclidean distance, is sequentially compact?
- ZW-SEQUENTIAL-COMPACTNESS-METRIC-SPACES-006 Let x_n=cos \!big(tfracnπ2big) in [-1,1]. Which values are limits of some subsequence?
- ZW-SEQUENTIAL-COMPACTNESS-METRIC-SPACES-008 For a metric space X, which statement is true?
- ZW-SEQUENTIAL-COMPACTNESS-METRIC-SPACES-009 Let K be a sequentially compact metric space and F⊆ K closed. Is F sequentially compact?
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