Let $a_0$ be any real number such that $0\lt a_0\lt1$. Define the sequence $a_1, a_2, a_3, ...$ by $a_{n+1} = \sqrt{1-a_n}$ for $n = 0, 1, 2, ...$ Show that $$\lim_{n\to \infty} a_n = {\sqrt{5} -1\over 2}$$
It seems clear that $$a_n = \sqrt{1- \sqrt{1-...\sqrt{1- \sqrt{1-a_0}}}}$$ where there are $n$ many square roots ($n \gt 0$). It is also easy to show that $$\sqrt{1- \sqrt{1-\sqrt{1-...}}} = {\sqrt{5} -1\over 2}$$
The issue is that I do not understand how to show that the effect of the radical containing $a_0$ disappears as $n$ tends to infinity, resulting in the desired equality. I would also appreciate any alternative approach to proving that the limit is as stated (based on the context in which I encountered the problem, I think it is likely that a reformulation or different interpretation of the problem will yield a simple proof).