I am reading Lee's book Introduction to smooth manifolds 2nd edition chapter 9 the fundamental theorem on flows.
In the proof of the fundamental theorem on flows the author defines $t_0=\inf\{t\in\mathbb{R}:(t,p_0)\notin W\}$, and argues that since $(0,p_0)\in W$, we have $t_0>0$. (Page 213 the last paragraph.)
But I am thinking can there be some $t\in\mathbb{R}$ s.t. $t<0$ and $(t,p_0)\notin W$? It is not very clear to me why this can not happen.
More context on the theorem (you may ignore this part):
In the proof, the author wants to show that for any smooth vector field $V$ on a smooth manifold $M$, there exists a local flow $\theta$ generated by $V$.
In the proof, the author defines $\mathcal{D}$ as the flow domain (we still need to prove that it is actually a flow domain), and wants to show that $\mathcal{D}$ is open and that $\theta$ as previously defined in the book is smooth on $\mathcal{D}$. The author wants to show this by contradiction.
He defines $W\subseteq\mathcal{D}$ s.t. $\theta$ is smooth on $W$. And that for each $(t,p)\in W$ there exists a product neighborhood $J\times U\subset \mathbb{R}\times M$ s.t. $(t,p)\in J\times U\subset W$. Where $J$ is an open interval in $\mathbb{R}$ containing both $0$ and $t$.
Now the author wants to show by contradiction that $W=\mathcal{D}$.
My understanding is the same as yours: it should be $t_0=\sup\{t\in\mathbb R:(t,p_0)\in W\}$.