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I wonder if the following is true:

Let $\pi : E \rightarrow M$ be a vector bundle over a smooth manifold $M$, $U$ an open set of $M$, and $r \in \Gamma(U,E)$ a local smooth section defined on $U$. Then, for every point $p \in U$, there exists a neighborhood $V$ of $p$ in $M$, whose closure is completely contained in $U$, and a global section $s \in \Gamma(E)$ which coincides with $r$ in $V$.

My idea is to reduce the support of $r$ to a compact set, and then use the pasting lemma for open subsets to extend trivially (by $0$) to a global section, but I couldn't concrete an argument.

Thank you in advance for any help.

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    $\begingroup$ As always, think bump function and/or partition of unity. $\endgroup$ May 27, 2023 at 23:29

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