A smooth exhaustion separates the locally finite chart bands

Choose a nonnegative smooth proper exhaustion $\rho$ from [L1]. Let $K_m:={\rho }^{-1}([m-1,m+2])(m\ge 1)\mathrm{.}$ Each $K_m$ is compact and the family $(K_m)$ covers $M$. If $m\equiv m^{\mathrm{\prime }}(\mathrm{m}\mathrm{o}\mathrm{d}4)$ and $m\mathrm{\ne }{m}^{\mathrm{\prime }}$, the defining intervals are separated by a positive gap.

Put $O_m:={\rho }^{-1}((m-5\mathrm{/}4,m+9\mathrm{/}4))$. Then $K_m\subseteq O_m$, and $O_m\cap O_{m^{\mathrm{\prime }}}=\mathrm{\varnothing }$ for distinct congruent indices modulo $4$. If $K_m=\mathrm{\varnothing }$, take $Q_m=1$ and $H_m=0$; all four requirements for this index are then immediate. Henceforth suppose $K_m\mathrm{\ne }\mathrm{\varnothing }$.

For every $p\in K_m$, a chart from [F1] can be shrunk over a Euclidean ball to a coordinate domain $(U,x)$ with $p\in U$, compact closure, and $\bar{U}\subseteq O_m$. Applying [L2] under countable choice to $\{p\}\subseteq U$ gives a smooth $\varphi :M\to [0,1]$ supported in $U$ and equal to $1$ on an open neighbourhood $V$ of $p$. The collection of all plateau neighbourhoods $V$ obtainable in this way covers $K_m$, so compactness gives a finite covering list of data $(U_{mj},x_{mj},{\varphi }_{mj},V_{mj})$, $1\le j\le r_m$. Countable choice selects one such list for each nonempty band $K_m$.

For each selected datum define a global block $B_{mj}:M\to {\mathbb{R}}^{n+1}$ by $B_{mj}(q):=\{\begin{array}{ll}({\varphi }_{mj}(q),{\varphi }_{mj}(q)x_{mj}(q)),& q\in U_{mj},\\ 0,& q\mathrm{\notin }{U}_{mj}\mathrm{.}\end{array}$ On the open cover $U_{mj}\cup (M\setminus \mathrm{supp}({\varphi }_{mj}))$ the two formulas are smooth and agree on the overlap, so [L3] makes $B_{mj}$ smooth. Set $H_m^{\mathrm{\prime }}:=(B_{m1},\dots ,B_{m{r}_m}):M\to {\mathbb{R}}^{r_m(n+1)}$. Its support lies in the finite union of the compact sets $\mathrm{supp}({\varphi }_{mj})\subseteq O_m$.

The map $H_m^{\mathrm{\prime }}$ separates points of $K_m$: if $H_m^{\mathrm{\prime }}(p)=H_m^{\mathrm{\prime }}(q)$, choose $j$ with $p\in V_{mj}$. Equality of the first coordinate of the $j$th block gives ${\varphi }_{mj}(q)=1$, and equality of the remaining coordinates gives $x_{mj}(p)=x_{mj}(q)$, whence $p=q$. It also separates tangent vectors: for $p\in V_{mj}$, the function ${\varphi }_{mj}$ is locally constant with value $1$, so the last $n$ components of $d{B}_{mj,p}$ are $d{x}_{mj,p}$, an isomorphism. Thus $d{H}_{m,p}^{\mathrm{\prime }}$ is injective for every $p\in K_m$.

Compact support makes $H_m^{\mathrm{\prime }}$ bounded. Choose $C_m\ge 1$ with $\mathrm{\parallel }{H}_m^{\mathrm{\prime }}(q)\mathrm{\parallel }\le C_m$ for every $q\in M$, and put $H_m:=2^{-m}{C}_m^{-1}{H}_m^{\mathrm{\prime }}\mathrm{.}$ This positive rescaling preserves support and both separation properties, and it gives $\mathrm{\parallel }{H}_m\mathrm{\parallel }\le 2^{-m}$ everywhere.

For a nonempty band, steps 4.1 and 6.1 put $\mathrm{supp}(H_m)$ inside $O_m$; for an empty band, step 2.1 gives empty support. The sets $O_m$ are disjoint for distinct congruent indices modulo $4$, so the corresponding supports are disjoint. Together with steps 1.1, 5.1, and 6.1, this proves all four stated properties.