Problem statement:
Determine the limit of the following sequence:
$\sqrt{a},\sqrt{1+\sqrt{a}}, \sqrt{1+\sqrt{1+\sqrt{a}}},... $
My progress:
Let´s begin by introducing some notation. Let $a_{n}$ denote the nth term of the sequence. We have $a_{1}=\sqrt{a}$ and $a_{n}=\sqrt{1+a_{n-1}}$. My instinct tells me now to rewrite as $a_{n}^2-a_{n-1}-1=0$ which has a root $\frac{1+\sqrt{5}}{2}$ (neglect the negative root for obvious reasons).
However: My friend told me this is only an eventually value of the sequence and not necessarily. I have to determine that this sequence converges before i can conclude this. How can I do this? And what does it actually mean when I solve the quadratic(because that is only an instinct of mine)?