Proof. Non-empty: any sequence \(\phi _k(x_0)\) in the compact \(K\) has a convergent subsequence. Compact: \(\omega (x_0) = \bigcap _{s\geq 0}\overline {\{\phi _t(x_0) : t \geq s\}}\), an intersection of compact sets. Invariant: if \(\phi _{t_k}(x_0) \to y\), then by continuity of the flow \(\phi _{t_k + t}(x_0) = \phi _t(\phi _{t_k}(x_0)) \to \phi _t(y)\), so \(\phi _t(y) \in \omega (x_0)\). Approach: if the distance did not tend to \(0\), a sequence \(\phi _{t_k}(x_0)\) would stay a fixed distance away from \(\omega (x_0)\), and a convergent subsequence would have its limit in \(\omega (x_0)\), a contradiction. Connectedness follows from approach, since a limit set split into two separated closed parts could not be approached by one continuous curve visiting both infinitely often. □

Proof. For \(x_0 \in K\), \(V(\phi _t(x_0))\) is non-increasing and bounded below on the compact \(K\), so it tends to a limit \(c\). If \(y \in \omega (x_0)\), then \(V(y) = \lim V(\phi _{t_k}(x_0)) = c\), so \(V = c\) on \(\omega (x_0)\). Since \(\omega (x_0)\) is invariant (Lemma 11.2.2), \(V(\phi _t(y)) = c\) for every \(t\), so \(\dot V(y) = 0\): \(\omega (x_0) \subseteq E\). Being invariant, it lies in \(M\). By the lemma, \(\phi _t(x_0)\) approaches \(\omega (x_0) \subseteq M\). □