Let $E\to M$ be a smooth vector bundle, let $(U_i)_{i\in I}$ be an open cover, and let $(\rho_i)_{i\in I}$ be smooth functions $\rho_i:M\to\mathbb{R}$ such that $\operatorname{supp}\rho_i\subset U_i$ for each $i\in I$ and the family $(\operatorname{supp}\rho_i)_{i\in I}$ is locally finite. For each $i\in I$, let $s_i\in \Gamma(E|_{U_i})$ be a smooth local section. If each product $\rho_i s_i$ is extended by zero outside $U_i$, then $s=\sum_{i\in I}\rho_i s_i$ defines a smooth global section of $E$.