On page 17 of the book Ordinary Differential Equations by Wolfgang Walter there is the following theorem pertinent to the initial value problem $$y'=f(x)g(y), y(\xi)=\eta$$
(H) $f(x)$ is continuous in an interval $J_x$; $g(y)$ is continuous in an interval $J_y$; and $\xi\in J_x$, $\eta \in J_y$. Theorem. Let hypothesis (H) from VII hold, let $g(\eta)=0$ and $g(y)\ne 0$ in an interval $\eta < y \le \eta + \alpha$ or $\eta-\alpha \le y <\eta, (\alpha>0)$, and let the improper integral
$$\int\limits_\eta^{\eta +\alpha} \frac{dz}{g(z)} \text{ or } \int\limits_{\eta-\alpha}^{\eta} \frac{dz}{g(z)},$$ respectively, be divergent. Then every solution that starts above or below the line $y=\eta$ remains (strictly) above or below this line (in both directions).
And then the following commentary follows:
It follows from this theorem that a solution $y(x)$ which satisfies $y>\eta$ at one point remains greater than $\eta$ for all $x$ (a corresponding statement holds for $<$). In particular, if $\eta$ is an interior point of $J_y$ and if both integrals diverge, then the initial value problem $y'=f(x)g(y), y(\xi)=\eta$ has only one solution $y(x)\equiv \eta$. This is the case, for example, if $g(y)$ has an isolated zero at the point $\eta$ and satisfies a Lipschitz condition at $\eta$
$$|g(y)-g(\eta)|=|g(y)|\le K|y-\eta|,$$
hence, in particular, if $g'(\eta)$ exists and is different from $0$.
I didn't understand from this theorem why, if a solution starts above (or below) $y\equiv\eta$ and remains above (or below) $y\equiv\eta$, it can't be one of the solutions of the ODE, and only $y\equiv\eta$ is possible? Also, how is the Lipschitz condition relevant here?
I would definitely appreciate if someone could please clarify these.