I am trying to solve the following problem from Lee's Introduction to Smooth Manifolds.
Exercise 4.37. Suppose $\pi:E\to M$ is a smooth covering map. Prove that every local section of $\pi$ is smooth.
My Attempt: Let $U\subset M$ be an open set and $\sigma: U\to E$ be a local section of $\pi$ i.e. $\pi \circ \sigma = \textrm{Id}_{U}$. For each $q\in U$, we prove that $q$ has a neighborhood such that $\sigma$ is smooth. Since $q\in M$, by definition of a smooth covering map, there is an evenly covered neighborhood $U_0$ of $q$ such that each component of $\pi^{-1}(U_0)$ is diffeomorphic to $U_0$.
Now, $\sigma(q)$ satisfies $\pi\circ \sigma(q)=q\in U_0$, so $\sigma(q)\in \pi^{-1}(U_0)$. Take the component $V_0$ such that $\sigma(q)\in V_0$. Then, $U_0$ is diffeomorphic to $V_0$, and therefore $U_0\cap U$ is also diffeomorphic to $V_0\cap \pi^{-1}(U)$ by $\pi$. In other words, the map $\pi|_{V_0\cap \pi^{-1}(U_0)}: V_0\cap \pi^{-1}(U_0)\to U\cap U_0$ is bijective.
This is where I am stuck. I feel like maybe I can use the uniqueness of an inverse to prove that $(\pi|_{V_0\cap \pi^{-1}(U_0)})^{-1} = \sigma|_{U_0\cap U}$, hence smooth, but I was unable to do so.
The following proposition was written right before this problem, so I might use it, but I couldn't figure out a way to use it.
Proposition 4.36. (Local Section Theorem for Smooth Covering Map) Suppose $E$ and $M$ are smooth manifolds with or without boundary, and $\pi:E\to M$ is a smooth covering map. Given any evenly open subset $U\subset M$, any $q\in U$ and $p$ in the fiber over $q$, there exists a unique smooth local section $\sigma: U\to E$ such that $\sigma(q)=p$.
You have to show that $\sigma$ is smooth. Since smoothness is a local property, it suffices to find for each $q \in U$ an open neighborhood $W$ which is contained in $U$ such that $\sigma \mid_W$ is smooth.
Let $W$ be an evenly covered (connected) open neighborhood of $q$ with $W \subset U$. The existence of such $W$ is shown in What is the relation of Lee's definition of a covering map to the standard definition?
Let $\tilde W$ be the component of $\pi^{-1}(W)$ which contains $\sigma(q)$. We know that $\pi \mid_\tilde W$ is a diffeomorphism. Let $\tau : W \to \tilde W$ denote the inverse diffeomorphism. We claim that $\sigma \mid_W = \tau$ which shows that $\sigma \mid_W$ is smooth.
Since $W$ is connected, $\sigma(W)$ is a connected set contained in $\pi^{-1}(W)$. Hence $\sigma(W)$ must be contained in a component of $\pi^{-1}(W)$. But we know that $\sigma(q) \in \tilde W$, thus $\sigma(W) \subset \tilde W$. Hence $\pi \mid_\tilde W(\sigma \mid_W(z)) = z$ for $z \in W$, thus $\sigma \mid_W(z) = (\pi \mid_\tilde W)^{-1}(z) = \tau(z)$.