Existence of partitions of unity on paracompact manifolds

Let $M$ be a paracompact smooth manifold and let $\{U_i\}_{i\in I}$ be an open cover of $M$. Then there is a family of smooth functions $\{\rho_i:M\to[0,1]\}_{i\in I}$ such that:

1. the supports $\operatorname{supp}(\rho_i)$ form a locally finite family;
2. $\operatorname{supp}(\rho_i)\subseteq U_i$ for every $i$; and
3. $\sum_{i\in I}\rho_i(x)=1$ for every $x\in M$.

Such a family is a smooth partition of unity subordinate to the cover.