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}).

I have two questions:

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?

Thanks!