2.1.11 Guillemin and Pollack

Let $X$ be a manifold with boundary. For each point $x\in \partial X$, we have a neighborhood in $X$, say $U_x$, which is diffeomorphic $\phi_x:U_x\to H^k$ to $k-$dimensional upper half space. Then let $\pi:U_x\to\mathbb{R}$ be projection onto the final coordinate (for $H^k$, this entry is $\geq 0$). Let $f_x=\pi\circ\phi_x$. We have constructed a collection of functions which, individually, are strictly positive on $X^\circ$ but $0$ on $\partial X$. By adding in the constant 1 function defined on $X^\circ$, how do I patch these together with a partition of unity? I know the definition of the partition of unity, but I don't know how to proceed.

I also am aware that we can refine our open cover $\lbrace U_x\rbrace\cup\lbrace X^\circ\rbrace$ to the countable one: $\lbrace U_{x_i}\rbrace\cup\lbrace X^\circ\rbrace$

Any help would be appreciated, thanks!

Let $\lbrace \theta_i\rbrace$ be a partition of unity on $X$ subordinate to the cover $\lbrace U_x\rbrace$. Then there are $\lbrace x_i\rbrace$ where $\text{supp}(\theta_i)\subset U_{x_i}$. Hence let each $f_{x_i}$ be extended to be zero off of $U_{x_i}$, toss in the constant 1 function $f_0$ on $U_{x_0}:= X^\circ$, and set $$f(x)=\sum_i \theta_if_{x_i}$$ Then it is clear that $f(x)>0$ on $x\in X^\circ$ and $f(x)=0$ for $x\in\partial X$. Then $f$ has $0$ as a regular value since each $f_{x_i}$ has $0$ as a regular value: for any $x\in\partial X$ we have
$$df_x=\sum_i d(\theta_if_{x_i})_x=\sum_i \theta_i(x)d(f_{x_i})_x+d(\theta_i)_xf_{x_i}(x)=\sum_i \theta_i(x)d(f_{x_i})_x$$
Evaluate at $\vec{n}(x)$ to see that $df_x$ is not identically zero