In an article on the Routh-Hurwitz test, I couldn't see why the following result is true:

Let $$p(s)=a_ns^n+a_{n-1}s^{n-1}+\dots+a_{1}s+a_0$$ and $$ \eta_{*} = \frac{a_n}{a_{n-1}}$$ $$g_{\eta}(s):=p(s)-\eta (a_{n-1}s^n+a_{n-3}s^{n-2}+\cdots)$$ Then one and only one zero of $g_{\eta}(s)$ tends to $\frac{-a_{n-1}}{a_n-\eta a_{n-1}}$ as $\eta \in \mathbb{R}$ approaches $\eta_{*}$ from the origin.

Notice that when $\eta=0$ then $g_{\eta}(s)=p(s)$ and when $\eta=\frac{s a_n}{a_{n-1}}$ then $$g(s)=p(s)-\frac{a_n}{a_{n-1}}(a_{n-1}s^n+a_{n-3}s^{n-2}+a_{n-5}s^{n-4}+\cdots)$$ a polynomial of degree $n-1$.

The title of the article is "Elementary proof of the Routh-Hurwitz test" (DOI link), and in the part quoted above, the author has already demonstrated that $p(s)$ and $g(s)$ have the same number of purely imaginary roots. I'd appreciate any help. Thanks in advance!