Smooth partitions of unity exist on manifolds

Apply [L1] and then [L2] to obtain an at-most-countable index set $I$, open sets $W_k$ and coordinate balls $V_k$ for $k\in I$ such that $M={\bigcup }_{k\in I}{W}_k$, the family $(V_k)$ is locally finite, and each $V_k$ lies in a selected cover member $U_{j(k)}$. The set $I$ is allowed to be finite or empty.

For each $k\in I$, [L3] gives a nonempty set of smooth functions $g_k:M\to [0,1]$ equal to $1$ near $\overline{W_k}$ and supported in $V_k$. Use [A1] to select one for each $k$. The family $(g_k{)}_{k\in I}$ is locally finite and pointwise positive because every point lies in some $W_k$. When $M$ is empty, $I$ may be empty and the partition claim is vacuous.

Normalize $(g_k)$ by [L4]; the resulting nonnegative smooth family $({\psi }_k{)}_{k\in I}$ has locally finite supports, sums to one, and satisfies $\mathrm{supp}({\psi }_k)\subseteq V_k\subseteq U_{j(k)}$ for every $k$.

To index the partition by the original cover as required by Smooth partitions of unity subordinate to an open cover, put ${\varphi }_j={\sum }_{k\in I:j(k)=j}{\psi }_k$ for each $j\in J$. Every sum is locally finite and smooth; the family $(\mathrm{supp}{\varphi }_j{)}_{j\in J}$ is locally finite, since near any point only finitely many ${\psi }_k$ have support. Nonnegativity and local finiteness give $\mathrm{supp}{\varphi }_j={\bigcup }_{k:j(k)=j}\mathrm{supp}{\psi }_k\subseteq U_j$, and ${\sum }_{j\in J}{\varphi }_j={\sum }_{k\in I}{\psi }_k=1$. Thus $({\varphi }_j{)}_{j\in J}$ is the required subordinate partition.