Question

Let (x_n) 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?

• A. If P is finite, then (x_n) has a monotonically increasing subsequence.
• B. If P is finite, then no subsequence of (x_n) is monotonically increasing.
• C. If P is infinite, then (x_n) has a monotonically decreasing subsequence.
• D. If P is infinite, then no subsequence of (x_n) is monotonically decreasing.

Answer 1

The Correct Option is A, C

To solve this problem, we must analyze the conditions for the set \( P \) and examine the behavior of the sequence \( (x_n) \). The problem gives us the conditions related to the set \( P \) and links them to the existence of monotonic subsequences. Let's explore each statement logically:

1. If \( P \) is finite, then \( (x_n) \) has a monotonically increasing subsequence:

This statement makes use of the Erdős–Szekeres theorem from combinatorics, which states that any sequence of real numbers has a monotonically increasing or decreasing subsequence, whose length depends on whether we consider a finite or infinite set \( P \). If \( P \) is finite, it implies that a large portion of the indices do not belong to \( P \). Thus, within these indices, a monotonically increasing subsequence must exist. Therefore, this statement is true.

1. If \( P \) is finite, then no subsequence of \( (x_n) \) is monotonically increasing:

This statement is false based on our explanation above. The finiteness of \( P \) ensures the existence of a monotonically increasing subsequence due to the complement of \( P \) being infinite.

1. If \( P \) is infinite, then \( (x_n) \) has a monotonically decreasing subsequence:

If \( P \) is infinite, there are infinitely many indices \( n \) such that \( x_n \) is larger than any \( x_m \) for \( m > n \). This makes the sequence behave in a manner where each of these terms forms the start of a decreasing subsequence by always finding larger terms, indicating the presence of a monotonically decreasing subsequence. Hence, this statement is true.

1. If \( P \) is infinite, then no subsequence of \( (x_n) \) is monotonically decreasing:

This statement contradicts the previous correct statement. If \( P \) is infinite, we are guaranteed the existence of a monotonically decreasing subsequence, which invalidates this assertion. Therefore, this statement is false.

Based on the analysis, the correct statements are:

• If \( P \) is finite, then \( (x_n) \) has a monotonically increasing subsequence.
• If \( P \) is infinite, then \( (x_n) \) has a monotonically decreasing subsequence.