Question:
Which of the following statements is/are TRUE?
Which of the following statements is/are TRUE?
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The Bolzano-Weierstrass theorem guarantees that every bounded sequence has a convergent subsequence. This is useful in analyzing the behavior of sequences and their subsequences.
Updated On: Jun 1, 2026
- There exists a monotone sequence that does not converge but has a convergent subsequence.
- There exists a sequence that has a bounded subsequence but does not have any convergent subsequence.
- There exists a sequence \( (x_n) \) such that given any positive integer \( m \), \( (x_n) \) has a subsequence converging to \( m \).
- There exists a sequence \( (x_n) \) such that \( (|x_{n+1} - x_n|) \) converges to 0 but \( (x_n) \) does not converge.
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The Correct Option is C, D
Solution and Explanation
Step 1: Analyze option (A).
A monotone sequence does not necessarily converge, but it may have a convergent subsequence.
For instance, the sequence \( x_n = (-1)^n \) does not converge, but the subsequence \( x_{2n} = 1 \) is convergent. Thus, option (A) is true.
Step 2: Analyze option (B).
A sequence cannot have a bounded subsequence but no convergent subsequence.
By the Bolzano-Weierstrass theorem, every bounded sequence has a convergent subsequence. Therefore, option (B) is false.
Step 3: Analyze option (C).
Consider a sequence where each term of the sequence equals the integer \( m \) repeated infinitely.
For any \( m \), such a sequence will have subsequences converging to \( m \), as each subsequence can be taken as the constant sequence of \( m \). Thus, option (C) is true.
Step 4: Analyze option (D).
This statement is true. A sequence like \( x_n = \sin \left(\frac{1}{n}\right) \) satisfies \( (|x_{n+1} - x_n|) \to 0 \) but does not converge.
The difference between successive terms approaches zero, but the sequence itself does not settle to a single value. Thus, option (D) is true.
Step 5: Conclusion.
From the analysis, we conclude that option (A) is the correct one, as it correctly describes the behavior of a monotone sequence that does not converge but has a convergent subsequence. Therefore, the correct answer is (A).
A monotone sequence does not necessarily converge, but it may have a convergent subsequence.
For instance, the sequence \( x_n = (-1)^n \) does not converge, but the subsequence \( x_{2n} = 1 \) is convergent. Thus, option (A) is true.
Step 2: Analyze option (B).
A sequence cannot have a bounded subsequence but no convergent subsequence.
By the Bolzano-Weierstrass theorem, every bounded sequence has a convergent subsequence. Therefore, option (B) is false.
Step 3: Analyze option (C).
Consider a sequence where each term of the sequence equals the integer \( m \) repeated infinitely.
For any \( m \), such a sequence will have subsequences converging to \( m \), as each subsequence can be taken as the constant sequence of \( m \). Thus, option (C) is true.
Step 4: Analyze option (D).
This statement is true. A sequence like \( x_n = \sin \left(\frac{1}{n}\right) \) satisfies \( (|x_{n+1} - x_n|) \to 0 \) but does not converge.
The difference between successive terms approaches zero, but the sequence itself does not settle to a single value. Thus, option (D) is true.
Step 5: Conclusion.
From the analysis, we conclude that option (A) is the correct one, as it correctly describes the behavior of a monotone sequence that does not converge but has a convergent subsequence. Therefore, the correct answer is (A).
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