MT【267】第一次很重要

\begin{equation*}
\textbf{已知}x_1,x_2<\pi,x_{n+1}=x_n+\left\{ \begin{aligned}
sin x_n &,x_n>x_{n+1}\\
cos x_n&,x_n\le x_{n+1}\\
\end{aligned} \right.
\end{equation*}
证明:$ x_n<\dfrac{3\pi}{2}$

假设存在$n_0,x_{n_0}<\dfrac{3\pi}{2},x_{n_0+1}\ge\dfrac{3\pi}{2},\because x_{n_0+1}-x_{n_0}\le1,\therefore x_{n_0}\ge x_{n_0+1}-1\ge \dfrac{3\pi}{2}-1>\pi,$

$\therefore \pi<x_{n_0}<\dfrac{3\pi}{2}$但此时由$x_{n_0+1}$ 的定义知道$x_{n_0+1}<x_{n_0}$ 与假设矛盾.所以假设不成立.