Smooth partitions of unity exist on manifolds with boundary

Form the set of all pairs $(B,j)$ where $j$ indexes a member $U_j$ of the given cover, $B$ is a coordinate ball or half-ball with compact closure, and $\bar{B}\subseteq U_j$. By [L1] the balls in these pairs cover $M$, without choosing one pair for every point. By [L2], their projection to balls has an at-most-countable subcover $(B_n)$. Use ${\mathrm{A}\mathrm{C}}_{\omega }$ to choose for each listed $B_n$ an index $j(n)$ witnessing $\overline{B_n}\subseteq U_{j(n)}$. If $M=\mathrm{\varnothing }$, the empty partition already satisfies [L6]; henceforth assume it is nonempty, repeating one selected pair if the subcover is finite.

Apply [L3] to $(B_n)$, obtaining an at-most-countable family $(W_k,V_k{)}_{k\in I}$ with ${\bigcup }_k{W}_k=M$, $\overline{W_k}\subseteq V_k\subseteq \overline{V_k}\subseteq B_{n(k)}$, and $(V_k)$ locally finite. Assign each $k$ to the original cover index $j(n(k))$ from step 1.1.

Fix $k$. A chart for $V_k$ identifies the compact set $\overline{W_k}\subseteq V_k$ with a compact subset $K_k$ of a relatively open set $Q_k\subseteq {\mathbb{H}}^n$. Choose an open $O_k\subseteq {\mathbb{R}}^n$ with $O_k\cap {\mathbb{H}}^n=Q_k$. For $n>0$, [L4] gives a smooth Euclidean bump equal to one near $K_k$ and supported inside $O_k$. Its support is compact by the finite-ball construction in [L4], so its intersection with the closed half-space is compact and lies inside $Q_k$. Restrict the bump to ${\mathbb{H}}^n$, pull it back through the chart, and extend by zero outside $V_k$. Its support is a compact subset of $V_k$, so this extension is smooth both at the manifold boundary and across the edge of $V_k$ by [L5]. It is a $[0,1]$-valued smooth function $b_k$, equal to one on $\overline{W_k}$ and supported inside $V_k$. For $n=0$, use the constant-one function on the singleton chart $V_k$, extended by zero outside it.

The nonempty set of valid bumps in step 3.1 is specified independently for each $k$ in an at-most-countable family. Use ${\mathrm{A}\mathrm{C}}_{\omega }$ once to select all $b_k$ (finite choice if $I$ is finite). Because $\mathrm{supp}{b}_k\subseteq V_k$ and $(V_k)$ is locally finite, $a:={\sum }_{k\in I}{b}_k$ is locally a finite smooth sum. It is positive everywhere because the $W_k$ cover $M$ and $b_k=1$ on $W_k$. By [L5], $1\mathrm{/}a$ is smooth in boundary charts: near each point a smooth local extension of $a$ stays positive after shrinking the chart, so the ordinary reciprocal composes smoothly. Set $c_k=b_k\mathrm{/}a$.

For each original cover index $j$, put ${\varphi }_j={\sum }_{k:j(n(k))=j}{c}_k$, taking the zero function when there are no such $k$. The family is locally finite because the $V_k$ are, and every ${\varphi }_j$ is smooth and nonnegative. The union of the locally finite closed supports assigned to $j$ is closed and lies in $U_j$, so $\mathrm{supp}{\varphi }_j\subseteq U_j$. Also ${\sum }_j{\varphi }_j={\sum }_k{c}_k=1$, and each ${\varphi }_j\le 1$. By [L6], this is a smooth partition subordinate to the original cover. The empty case was handled in step 1.1.