The following is an exercise I am doing for review for a midterm exam in differential geometry.
Question: Let $\pi:V\to X$ be any real differentiable vector bundle of rank $r$. Assume that $X$ is compact. Prove that there exists sections $\{\sigma_1,...,\sigma_N\}$ of $V$ over $X$ that generate each fiber (Remark that compactness is not necessary it just makes the argument easier).
Attempt
Let $\{U_j\}_{j\in J}$ be an open coordinate cover of $X$ and let $\{h_j\}_{j\in J}$ be local trivializations ($h_j:\pi^{-1}(U_j)\to U_j\times \mathbb{R}^r$ diffeomorphisms). Since $X$ is compact, we can find a finite subcover of $\{U_j\}_{j\in J}$ , say $\{U_1,...,U_k\}$. Define $\sigma_i^j(x)=h_j^{-1}(x,e_i)$. Then $\{\sigma_1^j(x),...,\sigma_r^j(x)\}$ defines a frame of $V$ over $U_j$, i.e. $\{\sigma_1^j(x),...,\sigma_r^j(x)\}$ is a basis for $V_x=\pi^{-1}(x)\enspace (*)$.
The idea at this point is not too complicated. Currently $\sigma_i^j:U_j\to V$ are local sections, and we want to extend them to global sections which vanish outside $U_j$. We can find a partition of unity $\{\varphi_j\}_{j=1}^k$ subordinate to the cover $\{U_j\}_{j=1}^k$ such that $supp(\varphi_j)\subseteq U_j$. Let $\gamma:X\to V$ denote the zero section. Set:
$$\overline{\sigma}_{i}^j(x):=\begin{cases} \varphi_j(x)\sigma_i^j (x) &if \enspace x\in U_j\\ \gamma(x) & if \enspace x\notin U_j \end{cases}.$$
Then $\overline{\sigma}_{i}^j:X\to V$ is a section. The next thing we want to show is $\{\overline{\sigma}_{i}^j\}_{1\leq j\leq k, 1\leq i\leq r}$ generate each fiber of $V$ over $X$. At first I thought this followed trivially by $(*)$ but now I am not sure. clearly for $x\in U_j,$ $\overline{\sigma}_{i}^j(x)\in Span(\sigma_i^j(x))$, but it is possible that $\varphi_j(x)=0$. Since $\sum_{j=1}^k\varphi_j(x)=1$, this can't be true for all $j$, there is definitely a $j_0$ such that $\varphi_{j_0}(x)>0$, but can $\overline{\sigma}_{i}^{j_0}(x)$ replace the role of $\sigma_{i}^j(x)$? I am kind of stumped at this point.
Any help is appreciated!