Compact Support on $\mathbb{R}^n$ Means Vanishing Outside a Bounded Set

Let $n\ge1$ be a natural number, let $\lVert\,\cdot\,\rVert$ be the Euclidean norm on Euclidean space $\mathbb{R}^n$, and let $d$ be the Euclidean distance, a metric on $\mathbb{R}^n$. By Metric Open Sets Form a Topology the subsets open in $(\mathbb{R}^n,d)$ form a topology on $\mathbb{R}^n$, and all topological notions below refer to it. Let $f:\mathbb{R}^n\to\mathbb{R}$.

1. (Vanishing off the support) $f(x)=0$ for every $x\in\mathbb{R}^n$ that does not lie in the support of $f$.

2. (Criterion) $f$ is compactly supported if and only if there is a real number $R>0$ such that $f(x)=0$ for every $x\in\mathbb{R}^n$ with $\lVert x\rVert>R$.

3. (Localisation) If $R>0$ is a real number such that $f(x)=0$ for every $x$ with $\lVert x\rVert>R$, then the support of $f$ is contained in the closed ball of centre $0$ and radius $R$ in $(\mathbb{R}^n,d)$.