The Wasserstein Distance between Two Probability Measures Carried by a Finite Set is Bounded by the Squared Diameter Times the Total Variation of the Masses

In the setting of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation, whose probability space $(\Omega,\mathcal{F},P)$ is not used, let $n,M\in\mathbb{N}$, let $z_{1},\dots,z_{M}\in\mathbb{R}^{n}$ be pairwise distinct and $F=\{z_{1},\dots,z_{M}\}$, whose subsets are Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §finite-sets, let $D\in\mathbb{R}$ satisfy $\lVert z_{i}-z_{j}\rVert\le D$ for all $i,j\in[M]$, and let $\rho,\rho'\in\mathcal{P}(\mathbb{R}^{n})$ satisfy $\rho(\mathbb{R}^{n}\setminus F)=0$ and $\rho'(\mathbb{R}^{n}\setminus F)=0$. The set $\mathcal{P}_{2}$ is that of The Second Moment of a Probability Measure on Euclidean Space and the Probability Measures with Finite Second Moment §space, $W_{2}$ the Wasserstein distance, and $\sum_{i=1}^{M}$ the finite sum.

(Bound)¶ $\rho,\rho'\in\mathcal{P}_{2}(\mathbb{R}^{n})$ and