I have a doubt about a theorem in Hirsch and Smale's Differential Equations, Dynamical Systems, and Linear Algebra textbook. In the chapter of Stability of equilibria they state
Theorem 2 Let $\bar{x}\in W$ be an equilibrium of the dynamical system $(1)$ and let $V:U\to \mathbb{R}$ be a Liapunov function for $\bar{x}$, $U$ a neighborhood of $\bar{x}$. let $P\subset U$ be a neighborhood of $\bar{x}$ which is closed in $W$. Suppose that $P$ is a positively invariant, and that there is no entire orbit in $P-\bar{x}$ on which $V$ is constant. Then $\bar{x}$ is asymptotically stable, and $P\subseteq B(\bar{x})$.
Before he writes
\begin{equation} x'=f(x)\tag{1} \end{equation} Where $f:W\to\mathbb{R}^n$ is a $C^1$ map on an open set $W\subseteq \mathbb{R}^n$.
and
The union of all the solution curves that tend toward $\bar{x}$ (as $t\to \infty$) is called the basin of $\bar{x}$, denoted by $B(\bar{x})$.
Now the beginning of the proof of Theorem 2 is
Imagine a trajectory $x(t)$, $0\le t<\infty$, in the positively invariant set $P$. Suppose $x(t)$ does not tend to $\bar{x}$ as $t\to\infty$. Then there must be a point $a\neq \bar{x}$ in $P$ and a sequence $t_n\to \infty$ such that \begin{equation} \lim_{n\to \infty}x(t_n)=a \end{equation}
So my question is: How do they conclude the existence of such a point $a$? The orbit $x(t)$ might tend to infinity as $t\to \infty$. That can be implied if $P$ is compact, but that is not in the hypothesis. The book Lições de Equações Diferenciais Ordinárias by Sotomayor has a similar theorem with $P$ closed only. Incidentally, a copy of Sotomayor's that I borrowed from the Library has the word compact written above $P$. If $P$ compact is indeed necessary, I would like to see a counterexample.