$X_n \to X$ in weakly, or in distribution iff $X_n-X\to 0$ weakly.
This seems to be true by Levy's Continuity Theorem. However, from the definition of weak convergence, i.e., $$\int f dP_{X_n} \to \int f dP_X \; \text{for all} \; f\in C_b(\mathbb{R}^d)$$ I don't see how the equivalence holds. I would greatly appreciate it if anyone could clear me up on this.
I need this result to show that $E(R\wedge |X_n-X|)\to 0$ below from the fact that $X_n \to X$ in distribution and $f(x)=R\wedge |x|$ is a bounded continuous function.
