# Fixed point iteration convergence

Was exploring the following fixed point iteration

$$x \leftarrow 1 + \frac{1}{c*log(1-x) - 1}$$

for $$x\in(0,1)$$ and $$c>1$$ and initial guess at

$$x_0 = \sqrt{1 - \frac{1}{c}}$$

It seems to always converge, but couldn't see how. If $$g(x)$$ is the RHS of above, the derivative goes to infinity as $$x \rightarrow 1$$, likewise the Lipschitz constant seems to also be infinite within the boundary.