Proof of Smooth Partitions of Unity on a Compact Smooth Manifold with Boundary

Write $\Omega_\alpha=\varphi_\alpha(U_\alpha)$ and let $H^n$ be the closed upper half-space, with $n$ the dimension of $M$.

Step 1 (local bump data). Fix $p\in M$ and a chart $(U_{\alpha(p)},\varphi_{\alpha(p)})$ with $p\in U_{\alpha(p)}$; write $\varphi=\varphi_{\alpha(p)}$, $\Omega=\Omega_{\alpha(p)}$, $z=\varphi(p)$. Since $\Omega$ is open in $H^n$, there is $s_p>0$ such that

$$K_p=\{x\in H^n: d(x,z)\le s_p\}\subseteq\Omega,$$

where $d$ is the Euclidean distance. Set $r_p=s_p/2$ and let $\chi^{(p)}:\mathbb{R}^n\to\mathbb{R}$ be a smooth bump function from Existence of Smooth Bump Functions on Euclidean Space with $\chi^{(p)}=1$ on the closed ball of radius $r_p$ around $z$ and $\chi^{(p)}=0$ at distance $\ge s_p$ from $z$, with values in $[0,1]$.

Define $\eta_p:M\to\mathbb{R}$ by $\eta_p(q)=\chi^{(p)}(\varphi(q))$ for $q\in U_{\alpha(p)}$ and $\eta_p(q)=0$ otherwise. I claim $\eta_p$ is a smooth differential $0$-form on $M$ (identified with a function as in that definition), with chart representatives $\eta_{p,\beta}=\eta_p\circ\varphi_\beta^{-1}$ on $\Omega_\beta$. Compatibility (property 3 with $k=0$) is immediate from this formula. For local smoothness (property 2) at $a\in\Omega_\beta$, let $q=\varphi_\beta^{-1}(a)$ and distinguish two cases. If $q\in U_{\alpha(p)}$: taking a local smooth extension $F$ of $\varphi\circ\varphi_\beta^{-1}$ at $a$ from Smooth Compatibility of Charts Modeled on the Closed Upper Half-Space, the function $\chi^{(p)}\circ F$ is smooth (iterated chain rule, as in the proof of Existence of Smooth Bump Functions on Euclidean Space) and agrees with $\eta_{p,\beta}$ near $a$ on $\Omega_\beta$. If $q\notin U_{\alpha(p)}$: then $q\notin\varphi^{-1}(K_p)$, and $\varphi^{-1}(K_p)$ is a closed subset of $M$ — indeed $K_p$ is closed and bounded in $\mathbb{R}^n$, hence compact by Heine-Borel Theorem in $\mathbb{R}^n$, so $\varphi^{-1}(K_p)$ is compact by Continuous Image of a Compact Space is Compact (applied to the continuous map $\varphi^{-1}$), and compact subsets of the Hausdorff space $M$ are closed (standard: for $q$ outside, separate $q$ from each point of the compact set and extract a finite subcover using Open Cover and Subcover of a Subset of a Topological Space). Hence there is an open neighborhood of $q$ in $M$ disjoint from $\varphi^{-1}(K_p)$, on which $\eta_p$ vanishes identically (outside $U_{\alpha(p)}$ by definition; inside because $\chi^{(p)}$ vanishes off $K_p$); so $\eta_{p,\beta}$ is locally the zero function near $a$. In both cases property 2 holds.

Moreover, the support of $\eta_p$ (as defined in the statement) is contained in $\varphi^{-1}(K_p)\subseteq U_{\alpha(p)}$: the set where $\eta_p\ne 0$ is contained in $\varphi^{-1}(K_p)$, which is closed, so every point outside it has a neighborhood meeting no nonzero values.

Step 2 (finite subcover). Let $V_p=\varphi_{\alpha(p)}^{-1}\bigl(\{x\in\Omega_{\alpha(p)}: d(x,\varphi_{\alpha(p)}(p))<r_p\}\bigr)$, an open subset of $M$ containing $p$ on which $\eta_p=1$. The family $(V_p)_{p\in M}$ is an open cover of $M$; since $M$ is compact, there are $N\in\mathbb{N}$ and points $p_1,\dots,p_N$ with $M=V_{p_1}\cup\cdots\cup V_{p_N}$. Set $\alpha_i=\alpha(p_i)$ and $\eta_i=\eta_{p_i}$.

Step 3 (normalization). Let $\sigma=\sum_{i=1}^{N}\eta_i$, a smooth $0$-form on $M$ (finite sums of smooth $0$-forms are smooth $0$-forms: chart representatives add, local smooth extensions add, and compatibility is preserved). For every $q\in M$ there is $i$ with $q\in V_{p_i}$, where $\eta_i(q)=1$; since all $\eta_j\ge 0$, we get $\sigma(q)\ge 1>0$. Define $\chi_i=\eta_i/\sigma$. In each chart, the representative of $\chi_i$ is the quotient of the representatives, whose local smooth extensions are quotients with denominator $\ge$ locally positive; these are smooth by Products and Quotients of C^k Real-Valued Maps on Euclidean Open Sets Are C^k. (To keep the denominator of the local extension positive, shrink the extension neighborhood using continuity of the extension of $\sigma$'s representative, which is $\ge 1$ on the chart image near the point.) Hence each $\chi_i$ is a smooth $0$-form on $M$.

Properties: $0\le\chi_i\le 1$ since $0\le\eta_i\le\sigma$ pointwise; the set where $\chi_i\ne 0$ equals the set where $\eta_i\ne 0$, so the support of $\chi_i$ equals that of $\eta_i$ and is contained in $U_{\alpha_i}$ by Step 1; and $\sum_{i=1}^{N}\chi_i(q)=\sigma(q)/\sigma(q)=1$ for every $q\in M$. $\blacksquare$