By the theorem 16.3 in this book, I know that $A$ as a union of open sets $A_V$, there exists a partition of unity on it. I denote it as $\{\phi_i\}$. Then for each $x\in A$, there exists an open neighborhood $U$ of it so that there's only finitely many support of $\phi$ intersects with $U$. This is the local finiteness. But I don't get why this guarantees there are only finitely many $\phi$ that are not identically zero on $M$ Any help on this? Thanks.
