I'm trying to understand the following theorem:
Theorem: Suppose that $x=x(t)$ is a solution of $\dot{x}=F(x)$ where $F$ is continuous. Suppose that $x(t)$ approaches a finite limit $\textbf{a}$ as $t \to \infty$. Then $\textbf{a}$ must be an equilibrium state for the equation, that is, $F(\textbf{a})=0$.
I found the following proof in the Sydsaeter and Hammond (last ed.) book:
but to me is not so intuitive. Can you explain it more easily and intuitively?