Let p>0 and $\displaystyle x = \sqrt{p+\sqrt{p+\sqrt{p+ \cdots }}}$ , where all the square roots are positive. Design a fixed point iteration $\displaystyle x_{n+1} = F (x_{n})$ with some F which has x as a fixed point. We prove that the fixed point iteration converges for all choices of initial guesses greater than -p+1/4.

Let $\displaystyle x_{n+1}=F(x_{n})= \sqrt{p+x_{n}}$ so x is a fixed point for F since F(x)=x.

Now let $\displaystyle g(x)=\sqrt{p+x}$. We have $\displaystyle g'(x)=\frac{1}{2 \sqrt{p+x}} $ .

I can see that for $\displaystyle x > -p + 1/4 $, we have that g'(x) <1.

From there I am not sure how to proceed to obtain convergence for $\displaystyle x_{0} > -p +\frac{1}{4} $ .