We know that if $(X_n)_n$ is a sequence of real random variables, then it converges in distribution to a random variable $X$ if and only if $\lim_nF_{X_n}(x)=F(x)$ at every point $x \in \mathbb{R}$ where $F_X$ is continuous.
I think the result is true if the variable $X_n$ takes values in $\mathbb{R^d}$, I mean $(X_n)_n=((X_n^1,...,X_n^d))_n$ converges in distribution to a random variable $X=(X_1,...,X_d)$ if and only if $\lim_nF_{X_n}(x_1,...,x_d)=P(X_n^1 \leq x_1,...,X_n^d \leq x_d)=F(x)$ at every point $(x_1,...,x_d) \in \mathbb{R^d}$ where $F_X$ is continuous.
But how can we prove it?