Let $X$ be Borel measurable and separable.

(i) $X_{n}\stackrel{\text{ as }}{\rightarrow}X$ and $d\left(X_{n},X\right)$ is asymptotically measurable;

(ii) $X_{n}\stackrel{\text{ as }}{\rightarrow}X$ and $X_{n}$ is asymptotically measurable.

How can we show that (i) implies (ii)?

(This is a part of Problem 1.9.1 on van der Vaart and Wellner, 1996)