I have problems with:
Let $a_1\in\mathbb{R}$ and $a_{n+1}=a_n^2-1$ for $\forall n\in\mathbb{N}$.
Prove, that if $|a_1|\leqslant \dfrac{1+\sqrt{5}}{2} $ then $a_n$ is limited, and otherwise,
if $|a_1|>\dfrac{1+\sqrt{5}}{2}$ then $a_n$ diverges to $+\infty$.
So I assumed that $a_{n+1}-a_n>0$. Then for sure $a_{n}\geqslant a_n^2-1$.
If we want $a_n$ to be increasing then it must lie between $\dfrac{1-\sqrt{5}}{2}$ and $\dfrac{1+\sqrt{5}}{2}$.
This is a place where problems appears. I am able to calculate it (to that point of course), but I don't fully understand what am I doing. Can somebody help me? Thanks in advance!