# smooth functions and partition unity

Take $U \subset \mathbb{R}^{n}$ where $U$ is bounded. Assume we have an open covering $U \subset \cup_{i=0}^{N}V_{i}$ where $V_{i} \subset U$ for each $i$. For each $V_{i}$ we have the mappings $v_{i} \in C^{\infty}(\bar{V_{i}})$. If we let $\{ \zeta_{i} \}_{i}^{N}$ be a smooth partition of unity subordinate to the open sets $\{V_{i}\}_{i=0}^{N}$ in $U$ and we define $v := \sum_{i=0}^{N}\zeta_{i}v_{i}$ then $v \in C^{\infty}(\bar{U})$.
Is it not required that each $\zeta_{i}v_{i}$ have $\bar{U}$ as a domain to define $v := \sum_{i=0}^{N}\zeta_{i}v_{i}$?
Secondly. how do we show that $v \in C^{\infty}(\bar{U})$, why is this true?