I am trying to prove the following lemma, that is used to prove the escape lemma (Lee's Intro to smooth manifolds Lemma 9.19)
Lemma
Suppose $X$ is a smooth vector field on a smooth manifold $M$ . Let $\gamma : J \to M$ a maximum integral curve of $X$ such that $b := $sup$(J)$ is finite. Let $t_0 \in J$, and $K \subseteq M$ compact. Suppose $\gamma([t_0, b)) \subseteq K$.
Suppose $U$ and $V$ are relatively compact open subsets of $M$ such that $K \subseteq U$ and $\bar U \subseteq V$ . Let $\psi \in C^\infty(M )$ such that $\psi|_ \bar U \equiv 1$ and supp$(\psi) ⊂ V$ .
Prove that there is a $\varepsilon > 0$ such that $(t_0 − \varepsilon, b) \subseteq J$ and $\gamma|_{(t_0−\varepsilon,b)}$ an is an integral curve of $\psi X$.
So I have no idea how to start, how can I prove this ?
The following proposition can be used
Proposition Let $X$ be a smooth vector field on a smooth manifold $M$ . Let $p \in M$, and $\gamma_p : J_p \to M$ the maximum integral curve of $X$ with $\gamma_p(0) = p$. Let $\gamma : J \to M$ be another integral curve of $X$ with $\gamma(0) = p$. Then $J \subseteq J_p$ and $\gamma = \gamma_p|_J$ .
Hint: First show that there is an $\epsilon>0$ such that $(t_0-\epsilon,t-0]\subset J$ and $\gamma(t_0-\epsilon,b)\subset U$. Then observe that by construction $X$ and $\psi X$ coincide on $U$ and use that $\gamma$ is an integral curve of $X$. (Maximality of $\gamma$ does not really play a role for the lemma.)