Let $\mathcal{M}^{+}(\mathbb{R}_{+})$ be space of non-negative Radon measures on $\mathbb{R}_{+}$ and define the metric $\rho$ on $\mathcal{M}^{+} (\mathbb{R}_{+})$ as $$ \rho(\mu,\nu)= \sup \left \{ \int_{\mathbb{R}_{+}} \psi d (\mu - \nu) ~|~ \psi \in C^{1}(\mathbb{R}_{+}), \|\psi \|_{\infty} \le 1 , \|\partial_{x} \psi \|_{\infty} \le 1 \right \} .$$ How to prove $\mathcal{M}^{+}(\mathbb{R}_{+})$ is complete w.r.t. $\rho$. I know that $$ \lim_{n \to \infty} \rho(\mu_{n},\mu) = 0 \iff \mu_{n} \to \mu~ \text{narrowly for}~ n \to \infty.$$ But how above equivalence can help us to prove the completeness ?