Consider the following statementI. Every Cauchy sequence contains convergent subsequenceII. If a subsequence of a Cauchy sequence converges to a real numbers l, then the original sequence also converge to l.III. Every monotone sequence contains a convergent subsequence.IV. Every bounded sequence contains a convergent subsequence.Select the correct answer using the codes given below:a)II, III and IVb)I, II and IVc)I, III and IVd)I, II, and IIICorrect answer is option 'B'. Can you explain this answer?

Question

Answer

Explanation:

To determine the correct answer, we need to evaluate each statement individually and see if it holds true.

I. Every Cauchy sequence contains a convergent subsequence:
This statement is true. A Cauchy sequence is a sequence in which the terms become arbitrarily close to each other as the sequence progresses. This means that for any positive tolerance, there exists a point in the sequence beyond which all the terms lie within that tolerance. This property allows us to select a subsequence that is convergent. By selecting terms from the original sequence that lie within each tolerance, we can construct a subsequence that converges to a specific limit.

II. If a subsequence of a Cauchy sequence converges to a real number l, then the original sequence also converges to l:
This statement is true. If a subsequence converges to a real number l, it means that all the terms beyond a certain index in the subsequence are arbitrarily close to l. Since the original sequence contains all the terms of the subsequence, it follows that all the terms beyond the same index in the original sequence are also arbitrarily close to l. Therefore, the original sequence also converges to l.

III. Every monotone sequence contains a convergent subsequence:
This statement is true. A monotone sequence is a sequence that is either non-decreasing or non-increasing. If the sequence is non-decreasing, we can choose a subsequence that consists of the terms at odd indices (1, 3, 5, ...), which is also non-decreasing. By the monotone convergence theorem, a bounded non-decreasing sequence must converge. Similarly, if the sequence is non-increasing, we can choose a subsequence that consists of the terms at even indices (2, 4, 6, ...), which is also non-increasing. Again, by the monotone convergence theorem, a bounded non-increasing sequence must converge. Therefore, every monotone sequence contains a convergent subsequence.

IV. Every bounded sequence contains a convergent subsequence:
This statement is true. A bounded sequence is a sequence in which all the terms lie within a certain range. By the Bolzano-Weierstrass theorem, any bounded sequence in a metric space has a convergent subsequence. Therefore, every bounded sequence contains a convergent subsequence.

Based on the evaluation of each statement, the correct answer is option 'B': I, II, and IV.

Answer

Correct answer should be C, counterexample for III: xn=n

Answer

Correct Answer will be B

Answer

The correct answer is 2: I, II, and IV.

Statement I: Every Cauchy sequence contains a convergent subsequence. This statement is true. Every Cauchy sequence is a type of sequence that converges to a specific value, and therefore it must contain at least one convergent subsequence.

Statement II: If a subsequence of a Cauchy sequence converges to a real number l, then the original sequence also converges to l. This statement is also true. If a subsequence of a Cauchy sequence converges to a specific value, then the entire sequence must also converge to that value, because the Cauchy sequence is defined as a sequence that converges to a specific value.

Statement III: Every monotone sequence contains a convergent subsequence. This statement is false. While it is true that every monotone sequence is bounded (that is, the values in the sequence are either all increasing or all decreasing, and therefore cannot go on indefinitely), not all bounded sequences contain convergent subsequences.

Statement IV: Every bounded sequence contains a convergent subsequence. This statement is true. Every bounded sequence is defined as a sequence whose values are all within a certain range, and therefore must contain at least one convergent subsequence.

Therefore, the correct answer is 2: I, II, and IV.

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