Existence of a Smooth Plateau Function on Euclidean Space

Adopt Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation and fix a dimension $q$. Smoothness of a function $\mathbb{R}^{q}\to\mathbb{R}$ is that of Smooth Functions on an Open Subset of ℝⁿ on $\mathbb{R}^{q}$, which is open by claim 1 of Euclidean Space Is Open in Itself. Let $x_{0}\in\mathbb{R}^{q}$ and let $r,s$ be real numbers with $r<s$.

Then there exists a smooth $\chi:\mathbb{R}^{q}\to\mathbb{R}$ such that $0\le\chi(x)\le1$ for every $x\in\mathbb{R}^{q}$, $\chi(x)=1$ whenever $\lVert x-x_{0}\rVert\le r$, and $\chi(x)=0$ whenever $\lVert x-x_{0}\rVert\ge s$.