Contraction mapping theorem for metric spaces - why is this part of the proof notable?

Question

Someone has made a remark to me that from the proof of the CMT for metric spaces we get that $$d(x, x_n) \leq \frac{K}{1-K}d(x_{n-1}, x_n).$$

What is this saying? Why is this remark worthy?

(The only thing I can think of is perhaps something to do with Lipschitz or continuity, but I can't see any obvious trails to this.)

Answer 1

It turns out that this is useful in some instances such as the following example:

Solve $x^7 - x^3 - 21x + 5 = 0$, for $x \in [0, 1]$, with error less than $10^{-6}$.

We can use the CMT and iteration to solve the equivalent equation $\frac{x^7 - x^3 + 5}{21} = x$. After some work and setting $f(x) = \frac{x^7 - x^3 + 5}{21}$ we can show that $f$ is a contraction map with $K = \frac{1}{3}$ and hence $\frac{K}{K-1} = \frac{1}{2}$.

Then, using the fact that $|x - x_n| \leq \frac{K}{K-1} |x_n - x_{n-1}|$, we see that all we need to do is iterate until two successive terms are within $2 \cdot 10^{-6}$ of each other (i.e. $|x_n - x_{n-1}| < 2 \cdot 10^{-6}$).

So $d(x, x_n) \leq \frac{K}{1-K}d(x_{n-1}, x_n)$ actually tells us when we can stop in this example.