Question

I don't understand iteration

Answer 1

Suppose you play a guessing game where someone thinks of a number, and then you try to guess it.  They tell you if your guess was too high or too low.

You can see that you can  eventually guess their number of any degree of precision.

Answer 2

Search "cobweb plot"

Answer 3

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.