A locally finite hat-function partition of unity on $\mathbb{R}$ subordinate to overlapping intervals

For $n\in\mathbb Z$, let $\psi(t)=\max\{1-|t|,0\}$ and $\varphi_n(x)=\psi(x-n)$. The functions are continuous, take values in $[0,1]$, and have support $[n-1,n+1]$. They are subordinate to the open cover $U_n=(n-\tfrac32,n+\tfrac32)$ of $\mathbb R$.

If $x\in[n,n+1]$, the only possibly nonzero functions are $\varphi_n$ and $\varphi_{n+1}$, and $\varphi_n(x)+\varphi_{n+1}(x)=(1-(x-n))+(1-(n+1-x))=1.$ Thus $\{\varphi_n\}_{n\in\mathbb Z}$ is a locally finite partition of unity subordinate to $\{U_n\}$. The support intervals show local finiteness directly: every bounded set meets only finitely many support intervals.