A sequence $(a\_n)$ is defined by $a\_n > 0$ and $a\_{n+1} = \sqrt{6+a\_n}$ for $n\ge1$. Show that

$1)$ the sequence $(a\_n)$ is monotone increasing if $0 < a\_1 < 3$

$2)$ the sequence $(a\_n)$ is monotone decreasing if $a\_1 > 3$

I know how to verify it by substituting values for $n$ but to prove generally, I don't know.
I know that the limit of the sequence is $3$ which is obtained by the positive root of $x^2 - x -6 =0$.

We usually use $a\_{n+1} - a\_n$ to check if a sequence is monotone increasing our monotone decreasing. But here, since intervals are involved, I don't know how to proceed.

Thank you in advance.