When does $f(a),f(f(a)),f(f(f(a)))...$ produce better and better approximations to $x=f(x)$?

Question

I tried to approximate the solution to $x=f(x)$ for some given $f$, by guessing $x=a$, then I observed that $x=f(a)$ was an even better approximation, and $x=f(f(a))$ and so on was even better, so why does this method work and for which f, is it sufficient that f is continuous?

Answer 1

This method certainly works if the function is Lipschitz continuous with Lipschitz constant $L<1$. In general, it is not true.

For example let $f(x)=x+1$, then for any starting value $x_0$, the sequence $x_n=f(x_{n-1})$ ($n\geq1$) diverges.