Let $U \subset \mathbb{R}^n$ be an open one and $X: U \rightarrow \mathbb{R}^n$ a gradient vector field of class $C^1$ with an amount (without an enumerable number of singularities). Given a trajectory $\psi(.; p)$ whose domain contains $\mathbb{R}^+$, show that $\omega (p)$ or is empty or is a unitary set.
The set $\omega (p)$ is : $\omega (p)=$ {$ q \in U; \exists \,\,\,\ \{ t_n \}; n \rightarrow \infty \Rightarrow ( t_n) \rightarrow \infty; \,\,\, \psi(t_n)=q $ }.
I'm trying to use the Poincaré Bendixson Theorem, but apparently it's not straightforward, because I need more hypotheses, for example, that the positive half-orbit was contained in a compact subset of $U$. I have tested many ways, but none seems to be correct . Someone can help!
As a gradient field, there is some function $f\in C^2$ so that $X=\nabla f$. Then for any solution $x$ of $\dot x=X(x)$ one has per chain rule $$ \frac{d}{dt} f(x(t))=f'(x(t))\dot x(t)=\|\nabla f(x(t))\|^2 $$ so that the value of $f$ is monotonically increasing along $f$. Now use that to conclude the claim on the $ω$ set.