Suppose $ω(Γ) = \{x^∗\}$ is a single element set where $Γ$ is an orbit of a locally Lipschitz vector field $f \colon E → \mathbb R^n$ with $E ⊂ \mathbb R^n$ is open.
Question: Show that for $x ∈ Γ$, we have $φ_t(x) → x^∗$ as $t → ∞$, where $φ$ is the flow. Conclude that $x^∗$ is an equilibrium point.
Hint: Assuming $φ_t(x)$ does not converge to $x^∗$, show that one can find $δ > 0$ such that the intersection of $Γ^+_x$ with the spheres $\{y∈\mathbb R^n :|y−x^∗|=δ\}$ is infinite.
The hint I was given confused me and not sure how to show the intersection of the orbit and sphere is infinite.
Proceed by contradiction: Assume that there is no convergence to $x^*$.
Then there exists $\delta>0$ such that the orbit $\varphi_t(x)$ intersects infinitely often the sphere of radius $\delta$ centered at $x^*$ (otherwise, for each $\delta$ the points $\varphi_t(x)$ would be inside the sphere for all sufficiently $t>T=T(\delta)$, but we are assuming that there is no convergence to $x^*$).
But since the sphere is compact there is an accumulation point and so a sequence $t_n\to+\infty$ such that that $\varphi_{t_n}(x)$ converges to some point $y$ on the sphere and so distinct from $x^*$. We know from the properties of the $\omega$-limit set that $y\in\omega(\Gamma)$, which gives a contradiction since $y\ne x^*$.
So there is convergence to $x^*$.