Let \(\left\{\mathbf{X}_{n}ight\}_{n=1}^{\infty}\) be a sequence of \(d\)-dimensional random vectors where \(\mathbf{X}_{n}\) has distribution function \(F_{n}\) for all \(n \in \mathbb{N}\) and let \(\mathbf{X}\) be a \(d\)-dimensional random vector with distribution function \(F\). Prove that if for any closed set of \(C \subset \mathbb{R}^{d}\),

\[\limsup _{n ightarrow \infty} P\left(\mathbf{X}_{n} \in Cight)=P(\mathbf{X} \in C),\]

then for any open set of \(G \subset \mathbb{R}^{d}\),

\[\liminf _{n ightarrow \infty} P\left(\mathbf{X}_{n} \in Gight)=P(\mathbf{X} \in G)\]