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Which of the following statements is FALSE?
- a)If a sequence X of real numbers converges to a real number and has two convergent subsequences X' and X" whose limits are not equal, then X is divergent.
- b)A Cauchy sequence of real numbers is unbounded.
- c)If a sequence (xn) of real number converges to a real number x, then any subsequence (xnK) of (xn) also converges to x.
- d)A bounded sequence of real numbers has a convergent subsequence.
Correct answer is option 'B'. Can you explain this answer?
Most Upvoted Answer
Which of the following statements is FALSE?a)If a sequence X of real n...
The statement (c) is FALSE.
In a sequence, if two subsequences converge to the same limit, it does not necessarily mean that the original sequence itself converges. It is possible for a sequence to have convergent subsequences without converging itself.
In a sequence, if two subsequences converge to the same limit, it does not necessarily mean that the original sequence itself converges. It is possible for a sequence to have convergent subsequences without converging itself.
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Which of the following statements is FALSE?a)If a sequence X of real n...
All the statements given are verified by different theorems,
Theorem: If a sequence converges then all subsequences converge and all convergent subsequences converge to the same limit (Option 3)
Theorem: Every bounded sequence has a convergent subsequence (Option 4)
Theorem: If {an}n∈N is a sequence that either has a subsequence that diverges or two convergent subsequences with different limits then {an}n∈N is divergent (Option 1)
Theorem:
1) A sequence {an} of real numbers is called a Cauchy sequence if for each ϵ > 0 there is a number N ∈ N so that if m, n > N then |an − am| < ϵ.
2) If a real sequence {an} converges, then for every ε > 0, there exists N ∈ N such that |an − am| < ε ∀ n,m ≥ N
3) Convergent sequences are Cauchy sequences.
A Cauchy sequence of real numbers is bounded (Option 2 is false)
A sequence is a convergent sequence if and only if it is a Cauchy sequence.
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Which of the following statements is FALSE?a)If a sequence X of real numbers converges to a real number and has two convergent subsequences X and X" whose limits are not equal, then X is divergent.b)A Cauchy sequence of real numbers is unbounded.c)If a sequence (xn) of real number converges to a real number x, then any subsequence (xnK) of (xn) also converges to x.d)A bounded sequence of real numbers has a convergent subsequence.Correct answer is option 'B'. Can you explain this answer? for Civil Engineering (CE) 2025 is part of Civil Engineering (CE) preparation. The Question and answers have been prepared according to the Civil Engineering (CE) exam syllabus. Information about Which of the following statements is FALSE?a)If a sequence X of real numbers converges to a real number and has two convergent subsequences X and X" whose limits are not equal, then X is divergent.b)A Cauchy sequence of real numbers is unbounded.c)If a sequence (xn) of real number converges to a real number x, then any subsequence (xnK) of (xn) also converges to x.d)A bounded sequence of real numbers has a convergent subsequence.Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for Civil Engineering (CE) 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Which of the following statements is FALSE?a)If a sequence X of real numbers converges to a real number and has two convergent subsequences X and X" whose limits are not equal, then X is divergent.b)A Cauchy sequence of real numbers is unbounded.c)If a sequence (xn) of real number converges to a real number x, then any subsequence (xnK) of (xn) also converges to x.d)A bounded sequence of real numbers has a convergent subsequence.Correct answer is option 'B'. Can you explain this answer?.
Which of the following statements is FALSE?a)If a sequence X of real numbers converges to a real number and has two convergent subsequences X and X" whose limits are not equal, then X is divergent.b)A Cauchy sequence of real numbers is unbounded.c)If a sequence (xn) of real number converges to a real number x, then any subsequence (xnK) of (xn) also converges to x.d)A bounded sequence of real numbers has a convergent subsequence.Correct answer is option 'B'. Can you explain this answer? for Civil Engineering (CE) 2025 is part of Civil Engineering (CE) preparation. The Question and answers have been prepared according to the Civil Engineering (CE) exam syllabus. Information about Which of the following statements is FALSE?a)If a sequence X of real numbers converges to a real number and has two convergent subsequences X and X" whose limits are not equal, then X is divergent.b)A Cauchy sequence of real numbers is unbounded.c)If a sequence (xn) of real number converges to a real number x, then any subsequence (xnK) of (xn) also converges to x.d)A bounded sequence of real numbers has a convergent subsequence.Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for Civil Engineering (CE) 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Which of the following statements is FALSE?a)If a sequence X of real numbers converges to a real number and has two convergent subsequences X and X" whose limits are not equal, then X is divergent.b)A Cauchy sequence of real numbers is unbounded.c)If a sequence (xn) of real number converges to a real number x, then any subsequence (xnK) of (xn) also converges to x.d)A bounded sequence of real numbers has a convergent subsequence.Correct answer is option 'B'. Can you explain this answer?.
Solutions for Which of the following statements is FALSE?a)If a sequence X of real numbers converges to a real number and has two convergent subsequences X and X" whose limits are not equal, then X is divergent.b)A Cauchy sequence of real numbers is unbounded.c)If a sequence (xn) of real number converges to a real number x, then any subsequence (xnK) of (xn) also converges to x.d)A bounded sequence of real numbers has a convergent subsequence.Correct answer is option 'B'. Can you explain this answer? in English & in Hindi are available as part of our courses for Civil Engineering (CE).
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ample number of questions to practice Which of the following statements is FALSE?a)If a sequence X of real numbers converges to a real number and has two convergent subsequences X and X" whose limits are not equal, then X is divergent.b)A Cauchy sequence of real numbers is unbounded.c)If a sequence (xn) of real number converges to a real number x, then any subsequence (xnK) of (xn) also converges to x.d)A bounded sequence of real numbers has a convergent subsequence.Correct answer is option 'B'. Can you explain this answer? tests, examples and also practice Civil Engineering (CE) tests.
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