I am trying to work out the (sketch of) proof of Ostrowski's Theorem (on classifying archimedean completely valued fields) in J.S. Milne's Algebraic Number Theory course notes (http://www.jmilne.org/math/CourseNotes/ANT.pdf, Remark 7.49, proof in footnote).
I will just recap the one spot I'm having trouble with: At this stage we have got a field $K$ complete with respect to an archimedean absolute value $\lvert \, \rvert$, with $\mathbb C\subseteq K$ and the restriction of $\lvert \, \rvert$ to $\mathbb C$ is the usual absolute value. We have taken an $x \in K\setminus \mathbb C$ and established the following inequality for all $z\in \mathbb C $ and $n \in \mathbb N$: $$\lvert x^n-z^n \rvert\geq \lvert x -z \rvert \lvert x \rvert^{n-1} .$$ The author then says ''on choosing $\lvert z \rvert <1$ and letting $n \to \infty$, we find that $\lvert x\rvert \geq \lvert x-z\rvert$''.
Now, I've divided both sides of the inequality above with $\lvert x \rvert^{n-1} $ and found $$ \lvert x \rvert + \lvert z \rvert \lvert z/x \rvert^{n-1} \geq \lvert x-z \rvert . $$
And indeed, we get what we want in the limit $n \to \infty$, if $ \lvert z \rvert< \lvert x \rvert$. But it is essential for the rest of the proof that we get this for all $\lvert z\rvert <1$.
Is there something I am missing here? I hope that I'm not overlooking something terribly obvious...