Inversions of a Permutation Form a Finite Set, and the Sign Map Exists

In the setting of Metric Spaces: Balls, the Metric Topology, and the Real and Complex Numbers as Metric Spaces, let $n\in\mathbb{N}$, with $S_{n}$ as in Permutations of a Set and the Set of Permutations of [n] §permutations, the set $\mathrm{Inv}(\sigma)$ of inversions of $\sigma\in S_{n}$ as in Inversions of a Permutation of [n] §inversions, finite sets and the number of elements $\#A$ of a finite set $A$, and, for $k\in\mathbb{N}_{0}$, the power $(-1)^{k}$ in $\mathbb{R}$ of the negative $-1$ of $1$, as in Powers with Exponents in the Natural Numbers with Zero §power and Powers with Exponents in the Natural Numbers with Zero §zero.

¶ For every $\sigma\in S_{n}$, the set $\mathrm{Inv}(\sigma)$ is finite.

¶ There is exactly one map $s:S_{n}\to\mathbb{R}$ with $s(\sigma)=(-1)^{\#\mathrm{Inv}(\sigma)}$ for every $\sigma\in S_{n}$, where $\mathrm{Inv}(\sigma)$ is finite by clause inversions.