I know that for any open cover $(\Omega_i)_{i\in I}$ of $\mathbb R$ we can find a $C^\infty$-partition of unity subordinated to $(\Omega_i)_{i\in I}$. Moreover, if $\eta$ is a mollifying kernel$^1$ on $\mathbb R$, then $$\eta_\varepsilon(x):=\frac1\varepsilon\eta\left(\frac x\varepsilon\right)\;\;\;\text{for }x\in\mathbb R^d$$ is in $C_c^\infty(\mathbb R)$ with $\operatorname{supp}\eta_\varepsilon\subseteq(-\varepsilon,\varepsilon)$.
Now I would like to construct something similar: I would like to obtain a family $(\rho_k)_{k\in\mathbb N}\subseteq C_c^\infty(\Omega)$ such that $\rho_k\equiv1$ on $\overline{\Omega_k}$, where $\Omega_k:=(-k,k)$, for all $k\in\mathbb N$ and $\rho_k\to1$ as $k\to\infty$. Or, more generally, for any open cover $(\Omega_i)_{i\in I}$ of $\mathbb R$.
Intuitively, this should be easy to obtain. $\rho_k$ simply needs to (rapidly and) smoothly decrease to $0$ at the endpoints of $(-k,k)$. But how can we construct $\rho_k$ (or, more generally, $\rho_i$) rigorously?
i.e. $\eta\in C_c^\infty(\mathbb R)$ with $\operatorname{supp}\eta\subseteq(-1,1)$, $\eta\ge1$, $\int\eta(x)\:{\rm d}x=1$ and $\eta(x)=\eta(y)$ for all $x,y\in\mathbb R$ with $|x|=|y|$.
There exists $\varphi \in C^\infty_c(\mathbb R)$ such that $\varphi>0$ on $(0,1),$ $\varphi = 0$ everywhere else, and $\int_0^1 \varphi =1.$
For $k=1,2,\dots $ define
$$\rho_k(x)= \int_{-\infty}^x [\varphi(t+k+1) - \varphi(t-k)]\,dt.$$
Then $\rho_k$ does the job, since $\rho_k=0$ on $(-\infty,-(k+1)],$ increases from $0$ to $1$ on $[-(k+1),-k],$ equals $1$ on $[-k,k],$ decreases from $1$ to $0$ on $[k,k+1],$ and equals $0$ on $[k+1,\infty).$