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 $\gamma = \gamma_p|_J$ .
Note I am reposting this question
How do I prove the escape lemma?
in which I mistankingly made people think I wanted any proof of the escape lemma, while my question is HOW TO PROVE (b), using the lemma(a) and the proposition, not how to prove the escape lemma in whatever way ignoring the whole post
Step b) is just basic general topology and does not use much of the Lemma. $\delta^{-1}(U)\subset\mathbb R$ is open, hence a disjoint union of open intervals, one of which is $J_U$. By definition $t_0\in J_U$ and $(t_0-\epsilon,b]\subset \delta^{-1}(U)$ and since $ (t_0-\epsilon,b]$ is connected and contains $t_0$, we must have $(t_0-\epsilon,b]\subset J_U$. This of course implies $[t_0,b]\subset J_U$.