Suppose that $x_0 \geq 2$ and $x_n = 2 + \sqrt{x_{n-1} - 2}$ for all natural $n$. Use the Monotone Convergence Theorem to prove that either $x_n \rightarrow 2$ or $x_n \rightarrow 3$ as $n$ grows.
Attempt: Suppose that $x_0 \geq 2$ and $x_n = 2 + \sqrt{x_{n-1} - 2}$ for all natural $n$. Then the Monotone Convergence Theorem says if the sequence is increasing and bounded above or decreasing and bounded below, then the sequence converges to a limit.
Then, when $x_0 = 2$ we have $x_n = 2$ for all $n$.
I don't know how to continue. I know if I find the limit of $x_n$ on both sides then we get $x = 2$ or $3$. But I don't know how to prove it. Please any feedback/hint or help will be appreciated. Thank you.