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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.

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