Exercise · ZW-SEQUENTIAL-COMPACTNESS-METRIC-SPACES-006 · ★★☆ Intermediate
Let \(x_n=\cos\!\big(\tfrac{n\pi}{2}\big)\) in \([-1,1]\). Which values are limits of some subsequence?
Choose one option.
Show answer
- B. \(\{-1,\,0,\,1\}\)
Explanation
The values of the sequence are \(0,-1,0,1\) repeated with period \(4\): each appears infinitely often, so each is the limit of a constant subsequence.
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-007 Let x_n=(-1)^n (n)/(n+1) in [-1,1]. To which values do its convergent subsequences converge?
- 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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