How to show that some sets tend to $0$ given a $X$-continuity set inside

Question

This is part of the proof for Theorem 3.8 from Billingsley's Convergence of Probability Measures.

Let $(X_n, X)$ be a random element of $S\times S$ and $\rho(X_n,X) \Rightarrow 0$ and if $A$ is an $X-$continuity set, then it follows that $P([X_n \in A]\Delta [X\in A])\to 0$.

The last bit of the proof comes down to showing that $P[\rho(X,A)<\epsilon, X\notin A]] + P[\rho(X,A^c)<\epsilon , X\in A]$ goes to 0 as $\epsilon \to 0$ if $A$ is an $X$-continuity set.

I can't figure out this part on how to show that the limits tend to zero if $P[X\in \partial A]=0$. I do get the pictures intuitively but I can't rigorously prove it. I would greatly appreciate some help and attach the full proof.

Answer 1

$P[\rho(X,A)<\epsilon, X\notin A]] \to P[\rho(X,A)=0, X\notin A]] \leq P(X \in \partial A)$ since $\rho(X,A)=0$ implies that $X \in \overline A$. For the second term note that $X \in A, X \in \overline {A^{c}}$ also implies $X \in \partial A$.