I've been working on this task for hours now but I just can't prove that this recursive sequence is convergent.
$$ a_0=5 \\ a_{n+1}=\frac15(2+6a_n-{a_n}^2) $$
Feeding some values to the calculator, one can clearly see that this sequence converges to 2. I firstly wanted to show that the sequence is monotone, however when I try doing something like that:
$$ a_{n+1} \geq a_n $$ I end up at: $$ -1 \leq a_n \leq 2$$ which just confuses me even more.
How can I show that this recursive sequence converges to 2?
Thanks very much in advance to all who try to help me.
Edit:
As stated in some comments below, I just need to show additionally that $ a_{n+1} \leq 2 $. But when I plug in the $ a_{n+1} $ from above I get $ \frac15(2+6a_n-{a_n}^2) \leq 2 $ what leaves me stuck again.