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E.g., for the Wasserstein distance $W_1(P,Q)$ between two probability measures $P$ and $Q$ one has the following duality formula: \begin{equation} W_1(P,Q)=\sup\Big\{\Big|\int f\,dP-\int f\,dQ\Big|\colon \text{Lip}(f)\le1\Big\}, \end{equation} where Lip$(f)$ is the Lipschitz constant for a function $f$. It then trivially follows that $|\int f\,dP_\epsilon-\int f\,dP|\le\epsilon$ for any $1$-Lipschitz function $f$ if $W_1(P_\epsilon,P)\le\epsilon$.