Chapter 9 - Infinite Series - 9.2 Monotone Sequences - Exercises Set 9.2 - Page 613: 16

False

Let us take an example: $a_n=(n-300)^2$ is eventually increasing but $a_{200} \lt a_{100}$ We can see that the sequence may be decreasing for $n \lt 1000$, and it can be eventually increasing. Thus, the given statement is false.