We're given
$$P_n(x) = 1+2x+3x^2+\cdots +nx^{n-1} \tag{1}\label{eq1A}$$
With the first linked solution, there's a complex $z$ and integers $i \neq j$, with $P_i(z)=P_j(z)=0$ and $w = z^{-1} \neq 0, 1$. Thus, $z = w^{-1}$ so, using \eqref{eq1A} with $P_i(w^{-1})$ and multiplying both sides by $w^{-1}$, we get
$$\begin{equation}\begin{aligned}
0 & = \color{blue}{1 + 2w^{-1} + 3w^{-2} + \cdots + iw^{-(i-1)}} \\
& = (w^{-1} + 2w^{-2} + 3w^{-3} + \cdots + (i-1)w^{-(i-1)}) + iw^{-i} \\
& = (\color{blue}{1 + 2w^{-1} + \cdots + iw^{-(i-1)}}) - (1 + w^{-1} + \cdots + w^{-(i-1)}) + iw^{-i} \\
& = 0 - \frac{1-w^{-i}}{1-w^{-1}} + iw^{-i}
\end{aligned}\end{equation}\tag{2}\label{eq2A}$$
Using $n = i + 1$, this gives
$$\begin{equation}\begin{aligned}
\frac{1-w^{-i}}{1-w^{-1}} &= iw^{-i} \\
\frac{w^{i}-1}{1-w^{-1}} & = i \\
w^{i}-1 & = i(1- w^{-1}) \\
w^{i+1}-w & = i(w - 1) \\
w^{i+1} & = (i+1)w - (i + 1 - 1) \\
w^n & = nw - n + 1
\end{aligned}\end{equation}\tag{3}\label{eq3A}$$
A corresponding result applies for $P_j(w^{-1})$. Note another way to get this is to instead use, for $x \neq 1$, that
$$f_n(x) = 1 + x + x^2 + \cdots + x^n = \frac{x^{n+1}-1}{x - 1} \tag{4}\label{eq4A}$$
to then get
$$P_n(x) = \frac{df_n(x)}{dx} = \frac{nx^{n+1}-(n+1)x^n+1}{(x-1)^2} \tag{5}\label{eq5A}$$
Regarding Corollary $2$ of the second solution, we have
$$f(x) = a_0 + a_{1}x + \cdots + a_{n}x^{n} \tag{6}\label{eq6A}$$
Then the solution says to use $f(x/R)$, but it should be $f(Rx)$ instead, for this to then become
$$\begin{equation}\begin{aligned}
f(Rx) & = a_0 + (Ra_{1})x + \cdots + (R^{n}a_{n})x^{n} \\
g(x) & = b_0 + b_{1}x + \cdots + b_{n}x^{n}
\end{aligned}\end{equation}\tag{7}\label{eq7A}$$
where $g(x) = f(Rx)$ and
$$b_i = R^{i}a_i \; \forall \; 0 \le i \le n \tag{8}\label{eq8A}$$
Using $R = \max\{a_0/a_1,\ldots,a_{n-1}/a_{n}\}$, and \eqref{eq8A}, we then get for all $0 \le i \le n-1$ that
$$\begin{equation}\begin{aligned}
R & \ge a_{i}/a_{i+1} \\
Ra_{i+1} & \ge a_{i} \\
R^{i+1}a_{i+1} & \ge R^{i}a_{i} \\
b_{i+1} & \ge b_{i}
\end{aligned}\end{equation}\tag{9}\label{eq9A}$$
Since this meets the conditions of lemma $1$, all roots $z_1$ of $g(x)$ have $\lvert z_1 \rvert \le 1$. Thus, with $f(Rx)$, all of its corresponding roots $z = Rz_1$ have $\lvert z \rvert = \lvert Rz_1\rvert = R\lvert z_1\rvert \le R$.
Showing $r \le \lvert z \rvert$ can be done similarly, except $f(rx)$ should be used instead of $f(x/r)$. A similar procedure to \eqref{eq7A} and \eqref{eq8A} gives, with $x \neq 0$, that
$$f(rx) = h(x) = c_0 + c_{1}x + \cdots + c_{n}x^{n} = x^{n}(c_0(x^{-1})^{n} + c_1(x^{-1})^{n-1} + \cdots + c_{n}) \tag{10}\label{eq10A}$$
with $c_{i+1} \le c_{i} \; \forall \; 0 \le i \le n - 1$. As stated in the solution, the polynomial in reverse using $\frac{1}{x}$ gives a set of positive, non-decreasing coefficients, so lemma $1$ applies to give that all roots $\left\lvert \frac{1}{z_1}\right\rvert \le 1 \; \to \; \lvert z_1 \rvert \ge 1$. Thus, with $f(rx)$, we have the roots $z = rz_1$, so $\lvert z\lvert = \lvert rz_1\rvert = r\lvert z_1\rvert \ge r$.
