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?

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

Answer 1

Answer: (A), (C)

If P is finite, then (xn) has a monotonically increasing subsequence.

Answer 2

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.