Problem: Suppose $M$ is a compact smooth manifold and $V:J\times M \to TM$ a smooth time-dependent vector field on $M$. Show that the domain $\mathcal{E}$ of the time-dependent flow of $V$ is all of $J\times J \times M$. $J$ is an open interval.
This problem is from Lee's Smooth manifolds book,problem 9-20
What I've tried: I've tried going at it as in the time independent case: for all $t\in J$ there is some $\epsilon >0$ such that $(t-\epsilon, t+\epsilon)\times (t-\epsilon, t+\epsilon)\times M \subset \mathcal{E}$. But this doesn't seem to help the problem.
Found the solution. One has to look at a maximal integral curve of the induced vector field $J\times M\to T(J\times M)$ and apply the Escape lemma.