I'm kind of struggling with an exercise I found in a book about Poincaré-Bendixon theory and I would like some help. The exercise is precisely what I wrote on the title: I have to show that if $f\colon\mathbb{R}^2\to\mathbb{R}$ is $C^2$ and we consider the equation $x'=\nabla f(x)$, any nonempty $\omega$-limit set is a critical point for the system.
Thank you.
Fact. If $\nabla f(x_0)\ne 0$, then for every neighborhood $V$ of $x_0$ there is another neighborhood $U\subset V$ of $x_0$ such that the orbit passing through $x_0$ will not return to $U$ after exiting $V$.
The above fact rules out limit cycles and periodic orbits that do not reduce to a single point.
Proof. Denote $M=|\nabla f(x_0)|$. Shrinking $V$ if necessary we can ensure $$\frac{M}{2}\le |\nabla f|\le 2M \tag{1}$$ holds in $V$. Being open, $V$ contains some ball $B(x_0,r)$.
Let $x(t)$ be the orbit with $x(t_0)=x_0$. By the above, this orbit stays in $V$ as long as $|t-t_0|<\frac{r}{2M}$. On this interval, the composition $f(x(t))$ has derivative at least $M/2$. Therefore, by the time the orbit exits $V$, the value of $f$ will have increased by at least $r/4$.
Let $U$ be a neighborhood of $x_0$ in which $|f(x)-f(x_0)|<r/4$. Since $f$ does not decrease along the orbits, it follows that the orbit through $x_0$ will not revisit $U$ after exiting $V$.