Real Analysis Monotone Convergence Theorem Question

I am working on the following exercise for an introductory Real Analysis course.

Suppose that $x_0 \geq 2$ and $x_n = 2 + \sqrt{x_{n-1}-2}$ for $n \in \mathbb{N}$. Use the Monotone Convergence Theorem to prove that either $\lim_{n \to \infty}x_n=2$ or $3$ or $\infty$.

I need some help to move in the right direction. I'll start with what I know so far. My first approach was to break the question into different cases.

I've found that if $x_0=2$, then $x_n =2$ for all $n \in \mathbb{N}$. So, if this is the case, the series is neither decreasing nor increasing? Furthermore, I've found that if $x_0 = 3$ then $x_n =3$ for all $n \in \mathbb{N}$. For $2 < x_0 < 3$, the series is monotone increasing, and converges to $3$. For $x_0 >3$, the series is monotone decreasing and converges to $3$. You will notice none of the cases I've given result in $\lim_{n \to \infty}x_n= \infty$. I have not proven any of the above.

Now, the Monotone Convergence Theorem states, according to my textbook:

A monotone increasing sequence that is bounded above converges.

A monotone decreasing sequence that is bounded below converges.

Now, if I can show that the series is monotone increasing/decreasing and bounded above/below for the different cases, I believe I can use the following.

$$L=\lim_{n \to \infty} x_{n+1}=\lim_{n \to \infty} 2+\sqrt{x_n -2}=2 + \sqrt{\lim_{n \to \infty}x_n-2}=2+ \sqrt{L-2}$$

So,

$$L=2+\sqrt{L-2}$$

Which, solving for $L$, yields the solutions $L=2$ and $L=3$. This makes sense according to what I claimed earlier. Once again, I have not found under what conditions the series is divergent. Any help would be appreciated.

Also, any advice as to proving that the series is either monotone increasing and bounded above, or monotone decreasing and bounded below, under the different conditions for $x_0$, would be appreciated.