I know this statement makes sense if the sequence were simply both increasing and decreasing without being monotone, that is, $x_{n+1} \geq x_n$ and $x_{n+1} \leq x_n.$
Actually, I am now confused how we know $x_{n+1} \geq x_n$ and $x_{n+1} \leq x_n$ implies that $x_{n+1} = x_n$.