Question:
Let (xn) be a sequence of real numbers. Consider the set P = {n\(\isin\N:x_n\gt x_m\) for all \(m\isin\N\) with \(m\gt n\)}. Then which of the following is/are true?
Let (xn) be a sequence of real numbers. Consider the set P = {n\(\isin\N:x_n\gt x_m\) for all \(m\isin\N\) with \(m\gt n\)}. Then which of the following is/are true?
Updated On: Oct 1, 2024
- If P is finite, then (xn) has a monotonically increasing subsequence.
- If P is finite, then no subsequence of (xn) is monotonically increasing.
- If P is infinite, then (xn) has a monotonically decreasing subsequence.
- If P is infinite, then no subsequence of (xn) is monotonically decreasing.
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The Correct Option is A, C
Solution and Explanation
The correct option is (A): If P is finite, then (xn) has a monotonically increasing subsequence. and (C): If P is infinite, then (xn) has a monotonically decreasing subsequence.
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