Prove $\sqrt{s_n+1} = \frac{1}{2}(1+\sqrt{5})$ This is to prove how the limit of $s_n$ converges to $\frac{1}{2}(1+\sqrt{5})$.
Assume: $s_1 = 1$; for $n \geq 1$, $s_{n+1} = \sqrt{s_n + 1}$.
How to prove this converges to $\frac{1}{2}(1+\sqrt{5})$?

Consider applying the Contraction Mapping Theorem. Just check that the conditions for the theorem are satisfied (I'll leave those details to you).
We have $s_{n+1} = \sqrt{s_n + 1}$. Take $F: x \to \sqrt{x +1}$. By the Contraction Mapping Theorem we can conclude that $F$ has a unique fixed point $s$.
This gives us:
\begin{align*}
   \lim s_{n+1} =& \lim\sqrt{s_n +1} \\
   s =& \sqrt{s+1}  \\
   s^2 - s - 1 =& 0
\end{align*}
Then by the quadratic formula we have that our unique fixed point $s = \dfrac{1+\sqrt5}{2}$.
