Let $f$ and $g$ be differentiable on a domain $D$ and suppose that $\gamma$ is a simple closed contour whose inside is contained in D.
If $|f(z)-g(z)|<|f(z)|$ for all $z$ on $\gamma$, then $f$ and $g$ have the same number of zeros inside $\gamma$ (counted including their order).
I was reading an example of application of Rouché's Theorem, where Rouché's theorem was used to show that the polynomial $p(z)=z^7-5z^3+12$ has $0$ roots in $\{z:\mathbb{C}:|z|<1\}$.
What was done was:
Let $g(z)=z^7-5z^3+12$ and let $f(z)=12$. Then for $|z|=1$,
$|f(z)-g(z)|=|z^7-5z^3| \\ \le|z|^7+5|z| \\=1+5\\=6<12=|f(z)|$
Hence by Rouché's Theorem $p(z)=z^7-5z^3+12$ has $7$ roots in $\{z:\mathbb{C}:|z|<2\}$.
I was wondering, what is the purpose to do the step $\le|z|^7+5|z|$? Can't I just jump straight from $|f(z)-g(z)|=|z^7-5z^3| \\=|1-5|\\=4<12=|f(z)|?$
Secondly, $|f(z)-g(z)|=|-z^7+5z^3|$, is there a reason why they used $|f(z)-g(z)|=|z^7-5z^3|$?
Thirdly, what does it mean by "(counted including their order)"? (From the definition above)