Q:

Why can't I use induction here?

We solved the following exercise in class: Show the following sequence converges:

$a_{n+1}=\sqrt{2+a_n}$
$a_1=0$

At first we tried to use mathematical induction to show it's monotonically increasing, but then for some reason the teacher said that's not the correct way to do it, and showed algebraically the for all $n\in{N}$ $a_{n+1}>a_n$.
Why can't we use induction here?

A:

$1$. Show that $\sqrt{2} \leq a_n \leq 2$, $\forall n \geq 1$ using induction.
$2$. Now show that $a_{n+1} > a_n$ using induction making use of the fact the fact you have proved in $1$.

