Solution to autonomous differential equation with locally lipscitz function

As I was learning about the following theorem and its proof from the book Nonlinear Systems by H. K. Khalil, I encountered a difficulty in grasping some parts of the proof.

Theorem: Consider the scalar autonomous differential equation

begin{equation}
dot{y}=-alpha(y), y(t_0)=y_0,tag{1}
end{equation}

where $alpha$ is a locally Lipschitz class $kappa$ function defined on $[0,a)$. For all $0leq{y_0}<a$, this equation has a unique solution $y(t)$ defined for all $tgeq{t_0}$. Moreover,

begin{equation}
y(t)=sigma(y_0,t-t_0),tag{2}
end{equation}

where $sigma$ is a class $kappaell$ function defined on $[0,a)times[0,infty)$.

The proof goes as follows.

Since $alpha(.)$ is locally Lipschitz, the equation (1) has a unique solution $forall {y_0}geq{0}$. Because $dot{y}(t)<0$ whenever $y(t)>0$, the solution has the property that $y(t)leq{y_0}$ forall $tgeq{t_0}$. By integration we have,

begin{equation}
-int_{y_0}^{y} dfrac{dx}{alpha(x)}= int_{t_0}^{t} dtau.
end{equation}

Let b be any positive number less than $a$ and define $eta(y)=-int_{b}^{y}dfrac{dx}{alpha(x)}$. The function $eta(y)$ is strictly decreasing differentiable function on $(0,a)$. Moreover, $lim_{yto{0}}eta(y)=infty$. This limit follows from two facts.

First, the solution of the differential equation $y(t)to{0}$ as $ttoinfty$, since $dot{y}(t)<0$ whenever $y(t)>0$.

Second, the limit $y(t)to{0}$ can happen only asymptotically as $ttoinfty$; it cannot happen in finite time due to the uniqueness of the solution.

Here I do not quite understand the second fact (in italics) how the uniqueness of solution ensures that $y(t)$ goes to $0$ asymptotically as $ttoinfty$.

Any hints on this are greatly appreciated.