$|z^n|< \epsilon/4$ by taking $n > \log(4/\epsilon) / \log(1/|z|)$

Question

$f_n(z) - z = -2z^{n+1}/(2z^n+1)$.

For any given value of $z$ we can make $|z^n|< \epsilon/4$ by taking $n > \log(4/\epsilon) / \log(1/|z|)$.

How does taking $n$ greater than that number make $|z^n|< \epsilon/4$?

Answer 1

Note that whenever $n > \log(4/\epsilon)/\log(1/|z|)$,

Case $|z|<1$, ($\log(1/|z|)>0$)

$-n \log(|z|) > \log(4/\epsilon)$ Then $\log(|z|^{-n}) > \log(4/\epsilon)$ so that

$|z|^{-n} > 4/\epsilon \iff |z|^n < \epsilon/4$

Answer 2

It works for $|z|<1$, as for such $z$ function $|z|^n$ is decreasing with $n$.