Let $\xi_n, \xi, \eta_n, \eta$ be random vectors in $\mathbb{R}^{d}$ such that $\xi$ and $\eta$ are independent, $\xi_n \overset{d}{\to} \xi$ and $\eta_n \overset{d}{\to} \eta$. Denote by $C(F_{\xi})$ the set of points of continuity of distribution function $F_{\xi}$ of $\xi$. Let $C(F_{\eta})$ have an obvious meaning. Prove of disprove that if $$\mathbb{P} \Big( (\xi_n, \eta_n) \in (-\infty,x]\times (-\infty,y] \Big) \to \mathbb{P} \Big( (\xi, \eta) \in (-\infty,x]\times (-\infty,y] \Big) \quad (*)$$ for all $x \in C(F_{\xi})$ and $y \in C(F_{\eta})$ then $(\xi_n, \eta_n) \overset{d}{\to} (\xi, \eta)$. We suppose that $z \le x$ iff $z_i \le x_i$ for all $1 \le i \le d$. I suppose that the answer is positive, but I'm not sure that my proof is correct and moreover, it looks that there should be a more simple proof.
My attempt to make a simple proof (not successful).
We know that $\xi_n \overset{d}{\to} \xi$ so $$\mathbb{P} \Big( \xi_n \in (-\infty,x] \Big) \to \mathbb{P} \Big( \xi \in (-\infty,x] \Big)$$ for all $x \in C(F_{\xi})$. Hypothesis: if $(x,y) \in C(F_{(\xi,\eta)})$ then $x \in C(F_{\xi})$ and $y \in C(F_{\eta})$ . If the hypothesis is true then it follows from (*) that the answer in the original problem is positive. Unfortunatelly the hypothesis is not true: counterexample is $\xi_n = \xi = 1, \eta_n = \eta = 0$, $x = y = 0$. So it's not a solution.
Addition
Theorem 2.2 from Billingsley, "Convergence of probability measures" looks like the answer, if I am right, because we can take $$\mathcal{U} = \{ (a,b] \times (c,d] | (P(\xi = a) = P(\xi = b) = 0 = P(\eta = c) = P(\eta = d) = 0) \},$$ then $\mathcal{U}$ is a $\pi$-system and each open set in $\mathbb{R}^d \times \mathbb{R}^d$ is a countable union of sets from $\mathcal{U}$ so if $$\mathbb{P} \Big( (\xi_n, \eta_n) \in (a,b]\times (c,d] \Big) \to \mathbb{P} \Big( (\xi, \eta) \in (a,b]\times (c,d] \Big) \quad (**)$$ for all $(a,b]\times (c,d] \in U$ then $(*)$ is true. Hence it's sufficient to prove $(**)$ but it looks like it follows immediately from $(*)$. Am I right? And is there a more simple proof?
Some more facts about problem.
If $\xi_n$ and $\eta_n$ are independent then the answer in the original problem is positive.
Moreover, $(x,y) \in C(F_{(\xi,\eta)})$ iff $\mathbb{P} \Big( (\xi,\eta) \in \partial((-\infty,x] \times (-\infty,y]) \Big) = 0$, where $\partial X$ is a boundary of $X$. Hence if $(x,y) \in C(F_{(\xi,\eta)})$ then either $x \in C(F_{\xi})$ or $y \in C(F_{\eta})$.
Billingsley, "Convergence of probability measures", doesn't contain the theorem in the form I need.
Any help is appreciated.