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I want to show the following

let $E$ be a vector bundle over a base manifold $M$. Let $s$ be a section of $E$ that vanishes at $x_0\in M$. Show that there exists a finite number of section $\{s_i\}_{i\in I}$ and smooth functions $\{f_i\}_{i\in I}$ such that $s=\sum_{i\in I} f_i\cdot s_i$.

I don't see where the vanishing comes in to play, and I also don't see how to define such a thing globally.

On a local trivialization, where we have $\phi:\pi^{-1}(U)\rightarrow U \times \mathbb R^n$, we can take the $s_i(x)=e_i$ and $f_i(x)=e_i\cdot s(x)$. But this is only local.

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    $\begingroup$ The fact that $s$ vanishes at $x_0$ is irrelevant. See this answer. $\endgroup$
    – Jack Lee
    Dec 8, 2019 at 20:07
  • $\begingroup$ As written, the question is completely trivial since you can just let $s$ itself be one of the $s_i$. $\endgroup$
    – Eric Wofsey
    Dec 8, 2019 at 20:43
  • $\begingroup$ @JackLee this does answer my question $\endgroup$
    – samlanader
    Dec 8, 2019 at 21:14
  • $\begingroup$ @EricWofsey thanks for pointing this out, I didn't notice. I must have taken down the notes wrong $\endgroup$
    – samlanader
    Dec 8, 2019 at 21:15

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