Let $M$ be a paracompact Hausdorff smooth manifold, let $\pi:E\to M$ be a smooth real vector bundle, and let $V\subset U\subset M$ be open subsets such that $\overline V$ is compact and $\overline V\subset U$, where the closure is taken in $M$. If $s\in \Gamma(U,E|_U)$ is a smooth section of the restricted bundle $E|_U\to U$, then there exists a smooth global section $\tilde{s}\in \Gamma(M,E)$ such that $\tilde{s}=s$ on an open neighbourhood of $\overline V$ and $\operatorname{supp}(\tilde{s})\subset U$.