I don't understand iteration

In fact, strictly speaking it doesn't always work. Whether it does or does not depends on the behavior of the function sufficiently close to the point you are trying to find the value at. Namely, the condition for iteration to work is that the function must "contract" a suitably small neighborhood of the desired fixed point x\_0 around that neighborhood. That is, if we have an interval half-width epsilon, then for every value of x in $x\_0 - epsilon, x\_0 + epsilon$, f(x) e $x\_0 - epsilon/A, x\_0 + epsilon/A$ (i.e. a shrunken interval, shrunk by the factor A) for a *fixed* downscaling factor A > 1 that is *independent* of epsilon, *provided* epsilon is smaller than some threshold. For a differentiable function, this condition corresponds to having |f'(x\_0)| < 1.