I am trying to prove the the escape lemma (Lee's Intro to smooth manifolds Lemma 9.19) The proof can be broken down in three parts
a) First prove this lemma:
Lemma: $X$ 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$(\psi) ⊂ V$ .
Then 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 $\psi X$.
b) Let $\delta$ be the maximal integral curve of $\psi X$ so that $\delta(t_0)=\gamma(t_0)$. Let $J_U \subseteq \Bbb R$ the connected component of $\delta^{-1}(U)$ that contains $t_0$. Prove that $[t_0, b]\subseteq J_U$
c) Derive a contradiction
Suppose step a is proven, **How should I prove step b and derive a contradiction **
The following proposition can be used twice in the whole proof
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 $\gamma = \gamma_p|_J$ .
See page 70 of https://mtaylor.web.unc.edu/wp-content/uploads/sites/16915/2018/04/analmv.pdf. It shows that if $\Omega \subset \mathbb{R}^n$ is open and $K \subset \Omega$ is compact, then there is a time $t_K > 0$ such that for all $x \in K$, the integral curve passing through $x$ exists on the interval $[-t_K, t_K]$.
Now turn to the case where $\Omega$ is a manifold. Then for each $x \in K$, we use coordinates to obtain a time of existence $t_x$ for points in a neighborhood of $x$. Then by covering $K$ by finitely many neighborhoods, we get a uniform time of existence for all points in $K$.