(Complex analysis) Understanding proof of Liouville's theorem In my textbook, Liouville's theorem was proved by showing
\begin{equation}
\begin{aligned}
f(z)-f(0)&=\frac{1}{2\pi i}\int_{|w|=R}f(w)\bigg(\frac{1}{w-z}-\frac{1}{w}\bigg)dw \\
&=\frac{z}{2\pi i}\int_{|w|=R}\frac{f(w)}{w(w-z)}dw
\end{aligned}
\end{equation}
Estimating the integral shows that 
$$|f(z)-f(0)|\leq\frac{|z|MR}{R(R-|Z|)}=\frac{|z|M}{R-|z|},$$
where $M\geq|f(z)|$. Since for a fixed $z$, the right side approaches $0$ as $R\to\infty$, $f(z)=f(0)$ for every $z$, and $f$ is constant.
My trouble is understanding the step where they "estimate the integral". How did they get that inequality?
 A: First, we assume that $|z|<R$.  Then, we have
$$\begin{align}
\left|\frac{z}{2\pi i}\oint_{|w|=R}\frac{f(w)}{w(w-z)}\,dw\right|&=\frac{|z|}{2\pi } \left|\int_0^{2\pi}\frac{f(Re^{i\phi})}{Re^{i\phi}(Re^{i\phi}-z)}\,Re^{i\phi}\,d\phi\right|\tag 1\\\\
&\le \frac{|z|}{2\pi }\int_0^{2\pi}\left|\frac{f(Re^{i\phi})}{Re^{i\phi}-z}\right|\,d\phi \tag 2\\\\
&\le \frac{|z|}{2\pi }\int_0^{2\pi}\frac{\left|f(Re^{i\phi})\right|}{\left|\,|Re^{i\phi}|-|z|\,\right|}\,d\phi \tag 3\\\\
&=\frac{|z|}{2\pi }\int_0^{2\pi}\frac{\left|f(Re^{i\phi})\right|}{\left|\,R-|z|\,\right|}\,d\phi \tag 4\\\\
&=\frac{|z|}{2\pi }\int_0^{2\pi}\frac{\left|f(Re^{i\phi})\right|}{R-|z|}\,d\phi \tag 5\\\\
&\le \frac{|z|}{2\pi }\int_0^{2\pi}\frac{M}{R-|z|}\,d\phi\tag 6 \\\\
&=\frac{|z|}{2\pi }\frac{M}{R-|z|}\int_0^{2\pi}(1)\,d\phi\tag 7\\\\
&=\frac{M|z|}{R-|z|}\tag 8\\\\
\end{align}$$

NOTES:
In arriving at $(1)$, we parameterized the contour $|w|=R$ as $w=Re^{i\phi}$, $\phi \in [0,2\pi]$.
In going from $(1)$ to $(2)$, we used $\left|\int_C f(z)\,dz\right|\le \int_C \left|f(z)\right|\,|dz|$.
In going from $(2)$ to $(3)$, we used the triangle inequality $|z_2-z_1|\ge |\,|z_2|-|z_1|\,|$.
In going from $(3)$ to $(4)$ we simply noted that $|Re^{i\phi}|=R$.
In going from $(4)$ to $(5)$, we used the assumption that $|z|<R$.
In going from $(5)$ to $(6)$, we used the bound for $f(z)$ (i.e., $|f(z)|\le M$).
In going from $(6)$ to $(7)$, we noted that the integrand of $(6)$ is constant with respect to $\phi$.
In going from $(7)$ to $(8)$, we integrated $\int_0^{2\pi}(1)\,d\phi=2\pi$
