A monotone subsequence is a subsequence that is either increasing or decreasing, which we define as follows:
- A sequence (S) is increasing if [tex]S_n \geq S_m[/tex] whenever n>m.
- A sequence (S) is decreasing if [tex]S_n \leq S_m[/tex] whenever n>m.
Here, we can create a subsequence by taking every sixth term:
• When n = 6, [tex]S_n = \cos(6\pi /3) = \cos(2\pi) = 1[/tex]
• When n = 12, [tex]S_n = \cos(12 \pi /3) = \cos(4\pi) = 1[/tex],
and so on, and every such element will have a value of 1. This is a monotone subsequence.
