This is a problem from Lee's Introduction to Smooth Manifolds, problem 10 chapter 10, can you guys please help me verify my solutions? Thanks!
Lee, problem 10.10:
Suppose $M$ is a compact smooth manifold and $q:E\rightarrow M$ is a smooth vector bundle of rank $k$. Use transversality to prove that $E$ admits a smooth section $\sigma$ with the property: if $k>\text{dim }M$, then $\sigma$ is nowhere vanishing; whilte if $k\leq\text{dim }M$, then the set of points where $\sigma$ vanishes is a smooth compact codimension-$k$ submanifold of $M$. Use this to show that $M$ admits a smooth vector field with only finitely many singular points.
My proof: For each $p\in M$, let $U$ be a coordinate ball centered at $p$ and $B$ an open subset of $U$ whose closure is contained in $U$ as well; furthermore, we may assume the vector bundle $E\rightarrow M$ is trivial over $U$ with local trivializing map $\Phi: q^{-1}(U)\rightarrow U\times\mathbb{R}^k$. Let $B_1,...,B_n$ be a finite open subcover of $M$, corresponding to $p_1,...,p_n$, and local trivializing maps $\Phi_1,...,\Phi_n$ over $U_1,...,U_n$. Let $\phi_i:M\rightarrow\mathbb{R}$ be a smooth function that is 1 on $B_i$ and supported in $U_i$, by replacing $\phi_i$ by $\phi_i/(\sum_{i=1}^n\phi_i)$, we may assume that $\sum_{i=1}^n\phi_i=1$. Define a map $F:M\times(\mathbb{R}^k)^n\rightarrow E$ as follows: $$F(p,(a_1^1,...,a_k^1),...,(a_1^p,...,a_k^p))=\sum_{i=1}^n\Phi_i^{-1}(p,a_1^i,...,a_k^i)\phi_i,$$ where we define $\Phi_i^{-1}(p,a_1^i,...,a_k^i)$ to be $0$ if $p\notin U_i$. $F$ is smooth because it is smooth on each $U_i\times(\mathbb{R}^k)^n$.
I claim $F$ is a submersion: if $p\in B_j\subset U_j$ and $v\in T_pM$, let $\gamma:(-\epsilon,\epsilon)\rightarrow M\times(\mathbb{R}^k)^n$ be a smooth curve such that $\gamma(0)=p$, $\gamma'(0)=(v,0,...,0)$, then $(\Phi_j\circ F\circ\gamma)'(0)=(v,0,...,0)$ since $\sum_{i=1}^n\phi_i=1$. On the other hand, if for any $a\in\mathbb{R}$, the smooth curve $\tau:(-\epsilon,\epsilon)\rightarrow(\mathbb{R}^k)^n$ given by $\tau(\epsilon)=(p,a\epsilon,0,....0)$ satisfies $\tau(0)=p$ and $\tau'(0)=(0,a,0,...,0)$, and $(\Phi_i\circ\tau)'(0)=(0,a,0,...,0)$. We can do this for all $kn$ number of $\mathbb{R}$ coordinates of $M\times(\mathbb{R}^k)^n$. Since $B_i$'s cover $M$, this shows that $F$ is a submersion.
In particular $F$ intersects transversely with $M$ identified as an embedded submanifold of $E$. Hence by the parametric transversality theorem, there exists $w\in(\mathbb{R}^k)^n$ such that the map from $M$ to $E$ given by $\cdot\mapsto F(\cdot,w)$ intersects transversely with $M$. The map is a smooth section by construction. The rest of the proof then follows.