# Exercise verification in John Lee's Introduction to Smooth Manifolds, Part 1

Let $$\pi: E\to M$$ be a smooth vector bundle, and $$V$$ an open subset of $$E$$ with the property that $$V \cap E_p$$ is non empty and convex for all $$p\in M$$.

By "a section of $$V$$" we mean a (local or global) section of $$E$$ whose image lies in $$V$$.

Show that there exists a smooth global section of $$V$$.

My Argument

Let $$p\in M$$ and $$(\sigma_i^p:U_p\to E)_{i=1}^k$$ a smooth local frame with $$p \in U_p\subseteq M$$ open subset.

Since $$E_p \cap V\ne\emptyset$$ then $$v^i_p\sigma_i^p|_p\in V \cap E_p$$ for a certain $$v_p \in \mathbb{R}^k$$.

Define a smooth local section $$\sigma^p:U_p\to E, q\mapsto v^i_p\sigma^p_i|_q$$, and take $$(\sigma^p)^{-1}(V)$$ which is open in $$M$$.

Then $$\{(\sigma^p)^{-1}(V):p\in M\}$$ is an open cover of $$M$$. Take a smooth partition of unity subordinate to it, say $$(\psi_p:p\in M)$$.

Consider $$\sigma=\sum_p\psi_p\sigma^p$$. Then $$\sigma$$ is a smooth global section of $$E$$.

It remains to show that the image of $$\sigma$$ lies in $$V$$.

Let $$q\in M$$, then I have $$\sigma(q)=\psi_p(q)\sigma^p|_q$$ for finite indices $$p$$, and I know that $$\sigma^p|_q$$ lies in $$V$$ for each $$p$$. It follows that $$\sigma(q)=\psi_p(q)\sigma^p|_q$$ lies in $$V$$ by the convexity of $$V\cap E_q$$. $$\qquad\square$$

Is my proof correct? It is possible to simply it?

P.S: this is the first part of Problem 13.2 in Lee's book: "Introduction to smooth manifolds, 2 Edition"

Here Exercise verification in John Lee's Introduction to Smooth Manifolds, Part 2 is the link to part 2 of the question.