I am reading Ordinary Differential Equations and Dynamical Systems by G.Theschl, and I am confused about the definition of the $\omega_{\pm}-$limit set of a set $X \subseteq M$. The definition states that $\omega_{\pm}(X)$ is the set of all points $y \in M$ for which exists sequences $t_n \rightarrow \pm \infty$ and $x_n \in X$ with $\Phi(t_n,x_n) \rightarrow y$
Now, consider the system $\dot{x}=x(1-x^2)$, $\dot{y}=-y$.
Clearly, the $x-$direction has two stable fixed points at $x=\pm1$ and an unstable fixed point at $x=0$. It is also clear that the $y-$direction has one stable fixed point at $y=0$.
The author claims that $\omega_+(B_r(0))=[-1,1] \times\{0\} $, $r>0$.
It is clear to me that $(0,0),(-1,0),(1,0)$ are in $\omega_+(B_r(0))$.
However, I don't get why the whole interval $[-1,1]$ is also there.
For a given point in $[-1,1] \times \{0\}$, what would be the two sequences $t_n$ and $x_n$ that satisfy the definition?
I would really appreciate if someone can clarify this for me.
Thanks!