Metrizability of weak convergence by the bounded Lipschitz metric

Why is the weak convergence of probability measures on $\mathbb{R}$ with respect to bounded continuous test functions $C^0_b(\mathbb{R})$ metrizable by the bounded Lipschitz metric $$d(\mu, \nu) = \sup_{f \in \text{Lip}(\mathbb{R})} \Big | \int_{\mathbb{R}} f d \nu - \int_{\mathbb{R}} f d \mu \Big |$$ where $$\text{Lip}(\mathbb{R}) = \Big \{ f \in C_b(\mathbb{R}) : \sup_x |f(x) | \leq 1, \sup_{x \neq y} \frac{| f(x) - f(y) |}{|x-y|} \leq 1 \Big \}?$$ For those who would like a reference, this is invoked in the proof of the truncated version of Wigner's semicircle law in Anderson-Guionnet-Zeitouni's $\textit{Introduction to Random Matrices}$ and is cited in the appendix as part of Theorem C.8, though no proof is given there. If anyone could help me with this fact, I'd greatly appreciate it!

Solution 1:

There is a proof in Section 8.3 of Bogachev's Measure Theory.